# THE NUMBER OF FARTHEST POINTS

Save this PDF as:

Size: px
Start display at page:

## Transcription

1 THE NUMBER OF FARTHEST POINTS T. S. MOTzKIN, E. G. STRAUS, AND F. A. VALENTINE 1. Introduction. Consider a set S in a metric space E. For each point x, let y{x) denote a point of S which has maximum distance from x> and let Y(x) be the set of all y{x) with that property. It is our purpose here to study sets S for which certain restrictions are placed on the number of points in Y(x). In 2 we analyze those sets S in the Minkowski plane for which Y(x) has exactly one element for each x S. In 3 we characterize those sets in the Euclidean plane E 2 for which Y{x) has at least two elements for each x S. In order to achieve these ends we first establish some introductory results which hold in rather general spaces. DEFINITION 1. Let S be a set in a metric space. If S is contained in a sphere of radius r, then its r- convex hull is the intersection of all closed spheres of radius r which contain S. A set S is r- convex if it coincides with its r- convex hull [ 2, p. 128]. LEMMA 1. Let S be a set of diameter d in a linear metric space. Then for each x S the set Y{x) lies in the boundary of the d- convex hull of S. Proof. If Y(x) j4 0, choose any point y(x) Then S is contained in a sphere with center at x and with radius d(x, y), where d(x, y) denotes the distance from x to y. Since for x S we have d{x 9 γ) < d, there exists a point z on the ray γx such that the sphere with center z and with radius d = d(z, y) contains S. The point y is thus clearly on the boundary of the J-convex hull. NOTE. By virtue of Lemma 1, all results for compact S below will hold under the less restrictive assumption that S contain the intersection of its closure with the boundary of its d- convex hull. COROLLARY 1. Let S be a set in a linear metric space. Then for each x the set Y(x) is contained in the boundary of the convex hull of S. Received March 1, This work was performed on a National Bureau of Standards contract with the University of California, Los Angeles, and was sponsored (in part) by the Office of Scientific Research, U.S.A.F. Pacific J. Math. 3 (1953),

2 222 T. S. MOTZKIN, E. G. STRAUS, AND F. A. VALENTINE This is an immediate consequence of the fact that S is contained in the sphere with center x and radius d(x 9 y {x)), provided Y(x) 0. LEMMA 2. Suppose S is a set in a linear metric space y and let T be a set such that Y{x) Φ 0 for each x T, Then d{x, y(x)) is a continuous function of x on T. Proof. Since \d(x 9 z) d{u 9 z)\ < d(x, u), and since I max d(x, z ) - max d{u, z)\ = \d(x, y(x)) - d{u, y(u))\, z e S z e S we have \d(x, y(x)) ~ d{u 9 y{u))\ < if d{x 9 u) <. LEMMA 3. Let S be a compact set in a linear metric space. If Xi > x 9 then all limit points of the sequence \y{xi) } lie in Y{x). Proof. Let y t = y{x{) be a sequence of points. Let γ be a limit point of the sequence ί yι i. Then the continuity oi d(x f y(x)) implies that d(x, y)>^d{x, q) for all q C S. Hence we have y'. Y(x). LEMMA 4. Let S be a compact set in a linear metric space, and suppose y{x) is single- valued on a set T. Then y(x) is a continuous mapping of T into S. Proof. Since y(x) is single - valued, Lemma 3 implies that if xι > #, then y(xi) > y(x)> 2. Sets in M 2 on which y(x) is single-valued. Let M 2 be a two-dimensional Minkowski space [2, p. 23]. We restrict our attention here to connected sets S in M 2 (See 4 for remarks about disconnected sets.) THEOREM 1. Let S be a continuum (compact connected set) in M 2 If y(x) is single- valued on S 9 then the set sum is the entire boundary B of the convex hull of S; and this convex hull is d- convex, where d is the diameter of S. Proof. According to Corollary 1, we have Σ χ S Y{x)cB.

3 THE NUMBER OF FARTHEST POINTS 223 By Lemma 4, the mapping y(x) yields a continuous mapping of S into B. Now the only connected sets in a simple closed curve are: (1) a point, (2) a simple arc, (3) the whole closed curve. For cases (1) and (2), let then the mapping y(x) of A into itself must have a fixed point x 0 = y(x\$ ), so that ί x Q! = Y{x 0 ) = S, in which case the theorem is trivial. Thus A = B in all three cases. Moreover, since by Lemma 1 the set A ~ B lies in the boundary of of the (/-convex hull of 5, the boundary of the d- convex hull must coincide with B. Since there is no continuous mapping without fixed points of a closed twocell into itself, Lemma 2 and Theorem! imply that, for single-valued y(x) 9 the connected bounded set 5 must contain the entire boundary of its convex hull, but not all of the interior of that hull (unless S consists of a single point). It may suffice, in some cases, to delete one single point from the interior of a convex set; for instance, in the case of a circular disc in E 2 > the deletion of the center makes y(x) single-valued throughout. In the remaining theorems and lemmas we restrict our attention to sets in E 2 DEFINITION 2. By a normal to a convex curve C at a point x C C we mean a line perpendicular to a line of support to C at x. NOTATION. We designate a line of support at x by L(x), and the corresponding normal by N(x). Further, for a point y C 5, we let x(y) be a point in S such that y - y (x ), and let X (y) be the set of all x (y). THEOREM 2. Suppose S is the boundary of a compact convex set in E 29 and suppose y{x) is single-valued on S. Then: (1) The set X(y) consists of all points of intersection of the normals to S at y with S - y. If S has a tangent at y, then x(y) is single-valued and continuous at y, (2) The mapping x(y) is monotonic; that is, the order of x(y ι ), #(72)> x(y 3 ) on S has the same sense as thatofγ i9 y 2 > 73* Proof. (1) U x = x{y), then the circle with center x and radius d(x, y) contains S. Hence the tangent to this circle at the point y is also a line of support to 5, and the radius lies in a normal to S at y.

4 224 T. S. MOTZKIN, E. G. STRAUS, AND F. A. VALENTINE Now, let yι > y, yι S, and choose x{ = #(y t ). Then, due to the continuity of the mapping y(x), each limit point of {χ( \ is in X(y). Thus if S has a tangent at a pointy, then the mapping x(y) is one-to-one and continuous at y. To complete the proof of (1), suppose S has a corner at y. Then the farthest points of intersection from y of the normals at y with S fill out a closed subarc of S, which we denote by S^ the end-points of S t we denote by α, and u r There exists a sequence y; S with y; >y such that the normals to S at the y z are unique and approach the left normal at y. Hence, by the above, x(yι) converges to Up and hence Uj X(y)> Similarly, u τ λ(y). The three lines determined by Up u r, and y divide the plane into seven closed sets, and the arc S^ is contained in that unbounded one which has u,u r as part of its boundary. We denote that set by A. Since each of the two circles with centers Ui and u r which pass through y contains S, it follows by the law of cosines that γ(u) = y for all «C A. Hence S t C X(y). According to Theorem 1, the curve S contains no straight line-segment, and thus any normal to S intersects S in exactly two points. Hence the common part {S~S ι ) X{y) is the null set, so that S ι = X{y). (2) The above facts, together with the fact that each u 1 S is contained in some X(y), imply that the transformation x{y) maps connected sets into connected sets, even though the mapping need not be single-valued and therefore not necessarily continuous. The single-valuedness of y{x) implies that if Ύι J r 2> tnen ^(yi ) ^(y 2 ) = 0. If the transformation x{y) failed to be monotonic, it would have a fixed point y = x{y); but this is impossible unless S is a single point. Hence condition (2) must hold. COROLLARY 2. Suppose C is the boundary of a compact convex set S. Let (X β be a diameter of C, and let /V(CX, /3) designate the common normal to C through CX and β. Then y{x) is single-valued on C if and only if for every pair of points u, v C which lie on the same side of N( 0C, /3)> the normals N(u) and N(v) intersect at an interior point of S. Proof. First observe that, for any compact convex set S with (X j3 as a diameter, if x α β = 0, then x and y(x) must lie on opposite sides of /V(CX, β). To prove the necessity, observe that (X and β are involutory points in the sense that y(y(α)) = α andy(y(/3)) = β. Hence the necessity follows from the monotonicity of y(x) as described in

5 THE NUMBER OF FARTHEST POINTS 225 Theorem 2 To prove the sufficiency, first choose x C C -(C (X/3). Suppose y(x) is not single-valued, and choose u, v C Y(x) As mentioned above, y(%) and x lie on opposite sides of iv( (X, β). A circle with center x and radius d(x, u) is tangent to C at both M and v, and the normals /V(u) and N(v) intersect at x, which is not interior to S. Hence y {x) is single- valued for x C {C 01/3). By continuity it follows also that This completes the proof. y(α) = j8, y(j8) = α. In the following we shall extend the generalized notions of curvature described by Bonnesen and Fenchel [2, pp ]. Choose a point % C, where C is a closed convex curve together with a line of support L(x). The circle tangent to L (x) at x and passing through a point p C C - x must have its center z (p) on the normal N(x) to L(x) at x. Establish an order on N(x) in terms of the distance from x, and let where δix) is an arc of C containing x. We define four types of centers of curvature as follows: E s (x, SU)) = sup z(p) 1 E t (x 9 Six)) = inf z(p) P P p G 8{x) - x, E s {x) E {x,c), E (x) = E s 8 t t {x, C). E (x) = lim Eίx,δix)), EΛx) s lira, (*, S(x)). s Hx)-*x ' SU)-.* ' Clearly z (x)<,(*) < o («) < s (%) relative ton(x). DEFINITION 3. The sets Σ,E s {x), ΣE o (x), ΣE^x), and Σ^U) (Λ; ranges over C) are respectively called the superior evolute, the oω er evolute, the inner evolute, and the inferior evolute of S, and are denoted by s, Z o, "j, *^. THEOREM 3. Suppose C is the boundary of the compact convex set S C 2 // y(*) is single - valued on C, ίaew ίae superior evolute, and hence all four

6 226 T. S. MOTZKIN, E. G. STRAUS, AND F. A. VALENTINE evolutes, of C must be contained in S. Proof, Since y(x) is single-valued for each point x x G C, the proof of Theorem 2 implies that for any normal N(x ι ) f the set N(x ί ) (C-~x ί ) consists of a single point, denoted by x '. Choose p G C - x χ. Since d{x%p) < d{x%x x ) = d(x',y(x')) 9 it is clear that the perpendicular bisector B of the segment x ι p intersects the segment x x x '. Hence B -x χ x' = zip) G S. THEOREM 4. Suppose the inner evolute of the boundary C of the compact convex set S is contained in S C. Then y(x) is single-valued on C. Proof. Suppose there exists an x G C such that γ{x) is not single-valued. Choose u, v G Y(x) The circle with center x and radius d(x, u) contains S and is tangent to C at u and v. Hence the arc uv of C - x contains a point w of minimal distance from x. The circle with center x and radius d(x, w) is tangent to C at w, while a neighboring arc of w on C lies outside or on that circle. Hence C has a unique normal at w and Ejiw) > x> so that Ei(w) is on or outside C. Theorems 3 and 4 do not determine the single-valuedness of y(x) on S if t, E o, and E s lie in S and contain points of C. This situation can be described as follows: THEOREM 5. Let S be a compact convex set with boundary C such that E s [and hence each of the evolutes) of C lies in S. Then y(x) fails to be singlevalued on C if and only if there exists a point x G C which lies on E( 9 E O9 and E S9 and which is the center of a circular arc contained in C. A Proof, To prove sufficiency, suppose there exists a point x G C which is the center of a circular arc C\ C C 9 and suppose y(x) is single-valued on C. Then according to Theorem 2 the single-valuedness of y(x) implies x for each y G C ί, Hence C ί C Y(x) 9 a contradiction. CX(y) To prove necessity, assume y{x) is not single-valued on C, Choose u, v G Y(x), and let w be a nearest point to x of the arc C γ of C - x joining u and v. In the proof of Theorem 4 we saw that E{(w) > x; but since the evolutes are By "center of a circular arc" we mean the center of the circle to which the arc belongs.

7 THE NUMBER OF FARTHEST POINTS 227 in S, we have E.(w) = E o (w) = E s (w) = x. (Since E s is bounded, C can contain no straight line segments.) Hence the circle with center x and radius d(x, w) contains S. Thus d{x, w) > d(x, u). From the definition of w it now follows that d{x, z) = d{x, u) for each z C t. Hence C L Hence C x is circular arc in C with center at x. As seen earlier, if S is a simply connected set containing at least two points, then y{x) the following theorem. is not single-valued on S. The situation is described more fully in THEOREM 6. Let S be a compact convex set in E 2 with boundary C. Then y(z) is single-valued if z^e s {x) for all x C; and y{z) is not singlevalued if z - E s (x), z ^ E o (x) for some x C Proof Assume y(z) is not single-valued; then there exist distinct points u ζl Y (z) 9 v (ϋy {z) 9 and the circle with center z and radius d(z f u) contains S and is tangent to C at u and v. Hence E s (u) - E s (v) - z. Now suppose there exists an x C such that z E s (x), z 4- E o (x). Then, since C is compact, there exists a point u ^ x, u ζl C 9 such that d(z,u) = d ( z 9 x ) = d ( z, y ( z ) ). Hence u Y (z ), x Y (z ). Thus Theorem 6 is proved. A few remarks about the four evolutes may be desirable at this point. The inferior and superior centers of curvature, EΛx) and E s (x), are determined by properties in the large. In fact, Ej contains the set of centers of those circles which are in S and which are tangent to C at not less than two points. Similarly E s contains the sets of centers of those circles which contain C and which are tangent to C at not less than two points. Since a convex curve C has curvature almost everywhere, we have Ei {x) = E o (x) for almost all x C. Let us define E Ξ Σ,E.(x) E o (χ), (x ranges over C), where, as usual, Eι(x) E o (x) denotes a closed segment. The number of normals to C through a point x E 2 > as a function of x, is the same in each component of the complement of E* In the case where S is a compact convex set for which E is bounded, there are exactly two normals to C through each point x in the unbounded component of the complement of E (the

8 228 T. S. MOTZKIN, E. G. STRAUS, AND F. A. VALENTINE lines joining y to the nearest and farthest points on C). However, from each point y ql E on E^(E S ) there are at least four normals to C. [According to Theorem 6, there are at least two normals to the two or more points of tangency u, v of the inscribed (circumscribed) circle with center at y. In addition, there are lines joining y to nearest (farthest) points on each of the two arcs of C joining u and v. ] Thus E^ and E s do not intersect the unbounded component of E. These statements imply the following: THEOREM 7. Let C be the boundary of a compact convex set S C E 2. E s C S if and only if E o C S. Also E s C S - C if and only if E o C S - C. Then AN EXAMPLE. Consider the family of ellipses C(e ), b 2 x χ 2 + a 2 x 2 = n 2 b 2, a > b. If the eccentricity e satisfies the condition e < \/2/2, then y{x) is singlevalued on C(e). If e > \/2/2, then y(x) is not single-valued at # = (0, ± ί> ) In each case the inner and outer evolutes coincide; they form the familiar astroid with cusps at ξ = ( a χ, 0), η = ( - a χ, 0), τ = ( 0, b χ ) and p = ( 0, - b γ ), where a ί < a, and b ί < b for e < \/2/2 while b γ > b for e > \[2/2. The superior e volute E s is the closed line- segment p T, and E, is the closed line-segment ξη. If e 0, then y{x) is single-valued on the complement of the open segment pτ-p-τ. 3. Sets on which Y(x) contains at least two points. THEOREM 8. Let S C E 2 be a compact set of diameter d, and let D denote the set of end-points of diameters of S. If Y (x) has at least two elements for each x ζl D, then Y(x) consists of exactly two points for x D \$ and D con" tains a finite number of points. The d- convex hull of S coincides with the d- convex hull Of D. [Since the latter is a Reuleaux polygon (see below), D must contain an odd number of points.] Proof. Let Σ = \C(x)\ be the family of circular boundaries C(x) with centers x D and with radii d. Let x D; then Y(x) = C(x) D. Since diam Y(x) < diam S = c?,

9 THE NUMBER OF FARTHEST POINTS 229 there exists a smallest arc A of C{x) which contains Y(x) 9 and which has a length not exceeding πd/6. Let x x and x 2 be the end-points of A. If a circle C{x') Σ were to intersect A x x x 2 > then C{x') would separate x ι and x 2 since length A <: πd/6. But this contradicts the fact that 5C C{x'). For any x D, we have z = y(#) if and only if x - y( z). Hence every x D is a point of intersection of at least two circles of Σ. These facts imply that } / (rc) = i^;,^2}. 1 Define where K(x) is the closed circular disk with center x and with radius d. Then each* Z) lies in the interior of all K (x) C // except K(xχ) and K(x 2 ), where Y (x ) = {χ ί9 χ 2 J. Hence Λ; is a corner-point of the boundary of H. As above, let A ι and,4 2 be the smallest arcs of C{x x ) and C(x 2 ) containing Y{x t ) and y(% 2 )> respectively. We have shown that A x A 2 =\x\; and /4 t and A 2 are in the boundary of //. Thus x is an isolated corner of the boundary of H. Hence D contains a finite number of points, and by definition the boundary of H is the boundary of the d~ convex hull of D. It is clearly a Reuleaux polygon, that is, a convex circular polygon whose arcs have radii d, and whose vertices are the centers of these arcs [2, pp ]. Finally, each of the circles in Σ contains S, and hence S C H. COROLLARY 3. Let S be a set satisfying the conditions of Theorem 8. Then Y(x)cD for each x S. of//. This is an immediate consequence of the fact that D consists of the vertices THEOREM 9. Let S C E 2 be a compact set such that Y(x) has at least two elements for each x C S. Then S lies in the union of a finite number of linesegments. Moreover, if Y (x) has exactly two elements for each x S, then S cannot be connected. Proof. Since Y(x) CD for each x S, the fact that Y(x) has a least two elements implies that x lies on the perpendicular bisector of the line joining two elements of D. Thus S is a subset of the set obtained by taking the union of the intersections of these perpendicular bisectors with H.

10 230 T. S. MOTZKIN, E. G. STRAUS, AND F. A. VALENTINE Since the set H has at least three corners x i9 x 2 and x 3, let S( (i = 1, 3) consist of those x G S such that {%(, x 2 \c Y{x). Each set Sj is nonempty since S( contains the center of the smaller arc of H joining xι and x 2. From the continuity of d(x 9 y(x)), it follows that S; is closed. Hence if 5 is connected, then S x S 2 Φ 0 (since S is compact), and thus there exists an%' G S such that Y(%') D { x ί9 x 2 y x 3 1. This establishes the theorem. We also obtain the following result due to Bing [ l]. COROLLARY 4. Let S be a bounded set in E 2 containing at least two points > and having the property that with every two points x G S> y G S there exists a z G S such that the triangle x y z is equilateral. Then S is the set of vertices of an equilateral triangle. Proof. The closure 5 of 5 must also satisfy the hypothesis stated. Consider the set D of Theorem 8 relative to S. If x G D, and ί y, z } C Y{ x), then d( y, z) = d, so that x, y, z form the vertices of a Reuleaux polygon, and therefore by Theorem 8 we have D = ί x, y, z\. Now let u be the centroid of the triangle x 9 y, z. By Theorem 9, S is contained in the segments xu, yu 9 and zu. Suppose t>g (5 xu - x); then 7 (i;) = ί y, z!. But t>, y, z is not equilateral; hence S xu ~ x. Similarly, S yu y, S zu = z. Consequently, S = ί Λ;, y, z! 4. Remarks and problems. Several questions are raised by our theorems. (1) If we try to characterize disconnected sets in E 2 for which γ{x) is single-valued, we see that this condition is not very restrictive. In fact, given any set S which contains at least one point of the boundary of its r- convex hull H for some radius r, we can adjoin a single point z to S, such that z lies on an interior normal to // at a point of H S, and such that y(x) relative to S + { z \ is single-valued on S + { z!. (2) The characterization of connected sets S in E n (n > 2) for which y{x) is single-valued on S offers considerable difficulties. The mapping y(x) still yields a continuous map of S into the boundary of its convex hull, but it need no longer be an onto mapping. For example, the torus, both the solid and its surface, will have single-valued y{x) for suitable ratios of the two radii. The argument that a nontrivial compact S which contains no indecomposable continua cannot be simply- connected holds, however, regardless of dimension, since every continuous mapping of such a simply-connected set S into itself has fixed points [4]. (3) The generalization of the discussion of multivalued y(x) suggests the

11 THE NUMBER OF FARTHEST POINTS 231 following problem: Let 5 be a compact set in E n such that Y(x) has at least k elements for x C S. Does it follow that 5 lies in the union of a finite number of (n ~ k + 1 )-dimensional planes? (Note that in the case k = 1 this is no restriction, while for k > n + 1 there would be no sets S. ) Are there any sets for which k = n + 1? It seems likely that this generalization is false, since the argument which proved the finiteness of the set D in Theorem 8 fails for n > 2. In the case k > n, all points of D are vertices of their c?-conves hull. Thus in this case D must surely be denumerable. (4) Is it possible to generalize Corollary 4, as follows: If the bounded set S in E n contains at least two points; and if, for some k > 2, with every two points x, y ζl S there are k 1 points in S which together with x, y form the vertices of a regular k- simplex, does it follow that S is the set of vertices of a regular Z- simplex, where k <l <n? (5) Another question raised by Corollary 4 is the following: What are the sets (bounded sets, compact sets) S in E 2 which have the property that with every x, y ζl S there is a z 1 S such that x y z is an isosceles triangle with vertex z and prescribed verticle angle (X? For Ot < 77/3, a nontrivial set with the stated property obviously cannot be bounded. For Ct = 77/3, the question for the bounded case is answered by Corollary 4. For α > 77/3, there is a considerable variety of bounded sets, although none of them can be finite. In fact, for Cί > 77/3 every S must be dense in itself; and thus, if closed, it must be perfect. The case (X =77 has been discussed by J. W. Green and W. Gustin [3]; for closed sets S, this case characterizes convexity. An easy argument shows that for compact 5, and 77/3 < 0L<π/2, line-segment joining two farthest points of S must be contained in S. the entire It may also be worth remarking that if S has the foregoing property for an angle (X, then its complement has the same property for the angle 77 CC. Thus the case Cί = 77/2 is especially noteworthy, since in this case the class of all S with the stated property is closed under the operation of taking complements. (6) Finally, one should compare the theorems about Y(x) with those for M(x), where M(x) denotes the subset of S whose points have minimum distance from x. In particular the theorem of Motzkin [6, 7] (see also Jessen [5]) states that a closed set S is convex if and only if M(x) is a single point for all x. This theorem does not correspond to any of the results on Y (x) in 1. In fact, the

12 232 T. S. MOTZKIN, E. G. STRAUS, AND F. A. VALENTINF analogous assumption, concerning a (not necessarily closed) set S in E n, that y(x) be single-valued for all x, is satisfied if and only if S consists of a single point. REFERENCES 1. R. H. Bing, On equilateral distance, Amer. Math. Monthly 58 (1951), T. Bonnesen and W. Fenchel, Theorie der konvexen Korper, Julius Springer, Berlin, 1934; reprinted by Chelsea, N.Y., J. W. Green and W. Gustin, Quasiconvex sets, Canadian J. Math. 2 (1950), O. H. Hamilton, Fixed points under trans formations of continua which are not connected im kleinen, Trans. Amer. Math. Soc. 44 (1938), B. Jessen, Two theorems on convex point sets (Danish), Mat. Tidsskr. B (1940), T. S. Motzkin, Sur quelques proprietes caracteristiques des ensembles convexes, Rendiconti della Reale Academia dei Lincei, Classe di Scienze Fisiche, Matematiche e Naturali, 21(1935), , Sur quelques proprietes caracteristiques des ensembles bornes non convexes, Rendiconti della Reale Academia dei Lincei, Classe di Scienze Fisiche, Matematiche e Naturali, 21 (1935), UNIVERSITY OF CALIFORNIA, LOS ANGELES

### (x 1, y 1 ) = (x 2, y 2 ) if and only if x 1 = x 2 and y 1 = y 2.

1. Complex numbers A complex number z is defined as an ordered pair z = (x, y), where x and y are a pair of real numbers. In usual notation, we write z = x + iy, where i is a symbol. The operations of

### MAT-INF4110/MAT-INF9110 Mathematical optimization

MAT-INF4110/MAT-INF9110 Mathematical optimization Geir Dahl August 20, 2013 Convexity Part IV Chapter 4 Representation of convex sets different representations of convex sets, boundary polyhedra and polytopes:

### Lecture Notes in Advanced Calculus 1 (80315) Raz Kupferman Institute of Mathematics The Hebrew University

Lecture Notes in Advanced Calculus 1 (80315) Raz Kupferman Institute of Mathematics The Hebrew University February 7, 2007 2 Contents 1 Metric Spaces 1 1.1 Basic definitions...........................

### The Multi-Agent Rendezvous Problem - Part 1 The Synchronous Case

The Multi-Agent Rendezvous Problem - Part 1 The Synchronous Case J. Lin 800 Phillips Road MS:0128-30E Webster, NY 14580-90701 jie.lin@xeroxlabs.com 585-422-4305 A. S. Morse PO Box 208267 Yale University

### 1/12/05: sec 3.1 and my article: How good is the Lebesgue measure?, Math. Intelligencer 11(2) (1989),

Real Analysis 2, Math 651, Spring 2005 April 26, 2005 1 Real Analysis 2, Math 651, Spring 2005 Krzysztof Chris Ciesielski 1/12/05: sec 3.1 and my article: How good is the Lebesgue measure?, Math. Intelligencer

### Supplementary Notes for W. Rudin: Principles of Mathematical Analysis

Supplementary Notes for W. Rudin: Principles of Mathematical Analysis SIGURDUR HELGASON In 8.00B it is customary to cover Chapters 7 in Rudin s book. Experience shows that this requires careful planning

### QUESTION BANK ON STRAIGHT LINE AND CIRCLE

QUESTION BANK ON STRAIGHT LINE AND CIRCLE Select the correct alternative : (Only one is correct) Q. If the lines x + y + = 0 ; 4x + y + 4 = 0 and x + αy + β = 0, where α + β =, are concurrent then α =,

### DECOMPOSITIONS OF HOMOGENEOUS CONTINUA

PACIFIC JOURNAL OF MATHEMATICS Vol. 99, No. 1, 1982 DECOMPOSITIONS OF HOMOGENEOUS CONTINUA JAMES T. ROGERS, JR. The purpose of this paper is to present a general theory of decomposition of homogeneous

### Real Analysis Math 131AH Rudin, Chapter #1. Dominique Abdi

Real Analysis Math 3AH Rudin, Chapter # Dominique Abdi.. If r is rational (r 0) and x is irrational, prove that r + x and rx are irrational. Solution. Assume the contrary, that r+x and rx are rational.

### CHAPTER 7. Connectedness

CHAPTER 7 Connectedness 7.1. Connected topological spaces Definition 7.1. A topological space (X, T X ) is said to be connected if there is no continuous surjection f : X {0, 1} where the two point set

### Helly s Theorem with Applications in Combinatorial Geometry. Andrejs Treibergs. Wednesday, August 31, 2016

Undergraduate Colloquium: Helly s Theorem with Applications in Combinatorial Geometry Andrejs Treibergs University of Utah Wednesday, August 31, 2016 2. USAC Lecture on Helly s Theorem The URL for these

### ON THE CORE OF A GRAPHf

ON THE CORE OF A GRAPHf By FRANK HARARY and MICHAEL D. PLUMMER [Received 8 October 1965] 1. Introduction Let G be a graph. A set of points M is said to cover all the lines of G if every line of G has at

### XIV GEOMETRICAL OLYMPIAD IN HONOUR OF I.F.SHARYGIN The correspondence round. Solutions

XIV GEOMETRIL OLYMPI IN HONOUR OF I.F.SHRYGIN The correspondence round. Solutions 1. (L.Shteingarts, grade 8) Three circles lie inside a square. Each of them touches externally two remaining circles. lso

### MAT 3271: Selected solutions to problem set 7

MT 3271: Selected solutions to problem set 7 Chapter 3, Exercises: 16. Consider the Real ffine Plane (that is what the text means by the usual Euclidean model ), which is a model of incidence geometry.

### Real Analysis Notes. Thomas Goller

Real Analysis Notes Thomas Goller September 4, 2011 Contents 1 Abstract Measure Spaces 2 1.1 Basic Definitions........................... 2 1.2 Measurable Functions........................ 2 1.3 Integration..............................

### Computability of Koch Curve and Koch Island

Koch 6J@~\$H Koch Eg\$N7W;;2DG=@- {9@Lw y wd@ics.nara-wu.ac.jp 2OM85*;R z kawamura@e.ics.nara-wu.ac.jp y F NI=w;RBgXM}XIt>pJs2JX2J z F NI=w;RBgX?M4VJ82=8&5f2J 5MW Koch 6J@~\$O Euclid J?LL>e\$NE57?E*\$J

### On the Length of Lemniscates

On the Length of Lemniscates Alexandre Eremenko & Walter Hayman For a monic polynomial p of degree d, we write E(p) := {z : p(z) =1}. A conjecture of Erdős, Herzog and Piranian [4], repeated by Erdős in

### MATHS 730 FC Lecture Notes March 5, Introduction

1 INTRODUCTION MATHS 730 FC Lecture Notes March 5, 2014 1 Introduction Definition. If A, B are sets and there exists a bijection A B, they have the same cardinality, which we write as A, #A. If there exists

### Exercise Solutions to Functional Analysis

Exercise Solutions to Functional Analysis Note: References refer to M. Schechter, Principles of Functional Analysis Exersize that. Let φ,..., φ n be an orthonormal set in a Hilbert space H. Show n f n

### Devil s Staircase Rotation Number of Outer Billiard with Polygonal Invariant Curves

Devil s Staircase Rotation Number of Outer Billiard with Polygonal Invariant Curves Zijian Yao February 10, 2014 Abstract In this paper, we discuss rotation number on the invariant curve of a one parameter

### MA651 Topology. Lecture 10. Metric Spaces.

MA65 Topology. Lecture 0. Metric Spaces. This text is based on the following books: Topology by James Dugundgji Fundamental concepts of topology by Peter O Neil Linear Algebra and Analysis by Marc Zamansky

### y mx 25m 25 4 circle. Then the perpendicular distance of tangent from the centre (0, 0) is the radius. Since tangent

Mathematics. The sides AB, BC and CA of ABC have, 4 and 5 interior points respectively on them as shown in the figure. The number of triangles that can be formed using these interior points is () 80 ()

### Euclidean Models of the p-adic Integers

Euclidean Models of the p-adic Integers Scott Zinzer December 12, 2012 1 Introduction Much of our visual perception is based in what seems to be standard Euclidean space; we can easily imagine perfectly

### 1 Topology Definition of a topology Basis (Base) of a topology The subspace topology & the product topology on X Y 3

Index Page 1 Topology 2 1.1 Definition of a topology 2 1.2 Basis (Base) of a topology 2 1.3 The subspace topology & the product topology on X Y 3 1.4 Basic topology concepts: limit points, closed sets,

### Maths 212: Homework Solutions

Maths 212: Homework Solutions 1. The definition of A ensures that x π for all x A, so π is an upper bound of A. To show it is the least upper bound, suppose x < π and consider two cases. If x < 1, then

### Definitions, Axioms, Postulates, Propositions, and Theorems from Euclidean and Non-Euclidean Geometries by Marvin Jay Greenberg ( )

Definitions, Axioms, Postulates, Propositions, and Theorems from Euclidean and Non-Euclidean Geometries by Marvin Jay Greenberg (2005-02-16) Logic Rules (Greenberg): Logic Rule 1 Allowable justifications.

### (D) (A) Q.3 To which of the following circles, the line y x + 3 = 0 is normal at the point ? 2 (A) 2

CIRCLE [STRAIGHT OBJECTIVE TYPE] Q. The line x y + = 0 is tangent to the circle at the point (, 5) and the centre of the circles lies on x y = 4. The radius of the circle is (A) 3 5 (B) 5 3 (C) 5 (D) 5

### the neumann-cheeger constant of the jungle gym

the neumann-cheeger constant of the jungle gym Itai Benjamini Isaac Chavel Edgar A. Feldman Our jungle gyms are dimensional differentiable manifolds M, with preferred Riemannian metrics, associated to

### Economics 204 Fall 2012 Problem Set 3 Suggested Solutions

Economics 204 Fall 2012 Problem Set 3 Suggested Solutions 1. Give an example of each of the following (and prove that your example indeed works): (a) A complete metric space that is bounded but not compact.

### Hanoi Open Mathematical Competition 2017

Hanoi Open Mathematical Competition 2017 Junior Section Saturday, 4 March 2017 08h30-11h30 Important: Answer to all 15 questions. Write your answers on the answer sheets provided. For the multiple choice

### After taking the square and expanding, we get x + y 2 = (x + y) (x + y) = x 2 + 2x y + y 2, inequality in analysis, we obtain.

Lecture 1: August 25 Introduction. Topology grew out of certain questions in geometry and analysis about 100 years ago. As Wikipedia puts it, the motivating insight behind topology is that some geometric

### Extreme points of compact convex sets

Extreme points of compact convex sets In this chapter, we are going to show that compact convex sets are determined by a proper subset, the set of its extreme points. Let us start with the main definition.

### 1 Radon, Helly and Carathéodory theorems

Math 735: Algebraic Methods in Combinatorics Sep. 16, 2008 Scribe: Thành Nguyen In this lecture note, we describe some properties of convex sets and their connection with a more general model in topological

### Finite affine planes in projective spaces

Finite affine planes in projective spaces J. A.Thas H. Van Maldeghem Ghent University, Belgium {jat,hvm}@cage.ugent.be Abstract We classify all representations of an arbitrary affine plane A of order q

### Hilbert s Metric and Gromov Hyperbolicity

Hilbert s Metric and Gromov Hyperbolicity Andrew Altman May 13, 2014 1 1 HILBERT METRIC 2 1 Hilbert Metric The Hilbert metric is a distance function defined on a convex bounded subset of the n-dimensional

### Web Solutions for How to Read and Do Proofs

Web Solutions for How to Read and Do Proofs An Introduction to Mathematical Thought Processes Sixth Edition Daniel Solow Department of Operations Weatherhead School of Management Case Western Reserve University

### ON THE INVERSE FUNCTION THEOREM

PACIFIC JOURNAL OF MATHEMATICS Vol. 64, No 1, 1976 ON THE INVERSE FUNCTION THEOREM F. H. CLARKE The classical inverse function theorem gives conditions under which a C r function admits (locally) a C Γ

### 4 CONNECTED PROJECTIVE-PLANAR GRAPHS ARE HAMILTONIAN. Robin Thomas* Xingxing Yu**

4 CONNECTED PROJECTIVE-PLANAR GRAPHS ARE HAMILTONIAN Robin Thomas* Xingxing Yu** School of Mathematics Georgia Institute of Technology Atlanta, Georgia 30332, USA May 1991, revised 23 October 1993. Published

### Auerbach bases and minimal volume sufficient enlargements

Auerbach bases and minimal volume sufficient enlargements M. I. Ostrovskii January, 2009 Abstract. Let B Y denote the unit ball of a normed linear space Y. A symmetric, bounded, closed, convex set A in

### MAXIMAL INDEPENDENT COLLECTIONS OF CLOSED SETS

proceedings of the american mathematical society Volume 32, Number 2, April 1972 MAXIMAL INDEPENDENT COLLECTIONS OF CLOSED SETS HARVY LEE BAKER, JR. Abstract. A theorem is proved which implies that if

### Definitions, Axioms, Postulates, Propositions, and Theorems from Euclidean and Non-Euclidean Geometries by Marvin Jay Greenberg

Definitions, Axioms, Postulates, Propositions, and Theorems from Euclidean and Non-Euclidean Geometries by Marvin Jay Greenberg Undefined Terms: Point, Line, Incident, Between, Congruent. Incidence Axioms:

### Partial cubes: structures, characterizations, and constructions

Partial cubes: structures, characterizations, and constructions Sergei Ovchinnikov San Francisco State University, Mathematics Department, 1600 Holloway Ave., San Francisco, CA 94132 Abstract Partial cubes

### ON SUCCESSIVE RADII AND p-sums OF CONVEX BODIES

ON SUCCESSIVE RADII AND p-sums OF CONVEX BODIES BERNARDO GONZÁLEZ AND MARÍA A. HERNÁNDEZ CIFRE Abstract. We study the behavior of the so called successive inner and outer radii with respect to the p-sums

### FIXED POINT THEOREMS FOR POINT-TO-SET MAPPINGS AND THE SET OF FIXED POINTS

PACIFIC JOURNAL OF MATHEMATICS Vol. 42, No. 2, 1972 FIXED POINT THEOREMS FOR POINT-TO-SET MAPPINGS AND THE SET OF FIXED POINTS HWEI-MEI KO Let X be a Banach space and K be a nonempty convex weakly compact

### INDECOMPOSABILITY IN INVERSE LIMITS WITH SET-VALUED FUNCTIONS

INDECOMPOSABILITY IN INVERSE LIMITS WITH SET-VALUED FUNCTIONS JAMES P. KELLY AND JONATHAN MEDDAUGH Abstract. In this paper, we develop a sufficient condition for the inverse limit of upper semi-continuous

### Hr(X1) + Hr(X2) - Hr(Xl n X2) <- H,+,(Xi U X2),

~~~~~~~~A' A PROBLEM OF BING* BY R. L. WILDER DEPARTMENT OF MATHEMATICS, UNIVERSITY OF MICHIGAN, ANN ARBOR Communicated July 8, 1965 1. In a recent paper,' R. H. Bing utilized (and proved) a lemma to the

### The edge-density for K 2,t minors

The edge-density for K,t minors Maria Chudnovsky 1 Columbia University, New York, NY 1007 Bruce Reed McGill University, Montreal, QC Paul Seymour Princeton University, Princeton, NJ 08544 December 5 007;

### QUESTION BANK ON. CONIC SECTION (Parabola, Ellipse & Hyperbola)

QUESTION BANK ON CONIC SECTION (Parabola, Ellipse & Hyperbola) Question bank on Parabola, Ellipse & Hyperbola Select the correct alternative : (Only one is correct) Q. Two mutually perpendicular tangents

### Epiconvergence and ε-subgradients of Convex Functions

Journal of Convex Analysis Volume 1 (1994), No.1, 87 100 Epiconvergence and ε-subgradients of Convex Functions Andrei Verona Department of Mathematics, California State University Los Angeles, CA 90032,

### THE CONLEY ATTRACTORS OF AN ITERATED FUNCTION SYSTEM

Bull. Aust. Math. Soc. 88 (2013), 267 279 doi:10.1017/s0004972713000348 THE CONLEY ATTRACTORS OF AN ITERATED FUNCTION SYSTEM MICHAEL F. BARNSLEY and ANDREW VINCE (Received 15 August 2012; accepted 21 February

### ON DENSITY TOPOLOGIES WITH RESPECT

Journal of Applied Analysis Vol. 8, No. 2 (2002), pp. 201 219 ON DENSITY TOPOLOGIES WITH RESPECT TO INVARIANT σ-ideals J. HEJDUK Received June 13, 2001 and, in revised form, December 17, 2001 Abstract.

### WILSON ANGLES IN LINEAR NORMED SPACES

PACIFIC JOURNAL OF MATHEMATICS Vol. 36, No. 1, 1971 WILSON ANGLES IN LINEAR NORMED SPACES J. E. VALENTINE AND S. G. WAYMENT The purpose of this note is to give a complete answer to the question: which

### ON THE "BANG-BANG" CONTROL PROBLEM*

11 ON THE "BANG-BANG" CONTROL PROBLEM* BY R. BELLMAN, I. GLICKSBERG AND O. GROSS The RAND Corporation, Santa Monica, Calif. Summary. Let S be a physical system whose state at any time is described by an

### Convexity in R n. The following lemma will be needed in a while. Lemma 1 Let x E, u R n. If τ I(x, u), τ 0, define. f(x + τu) f(x). τ.

Convexity in R n Let E be a convex subset of R n. A function f : E (, ] is convex iff f(tx + (1 t)y) (1 t)f(x) + tf(y) x, y E, t [0, 1]. A similar definition holds in any vector space. A topology is needed

### (NON)UNIQUENESS OF MINIMIZERS IN THE LEAST GRADIENT PROBLEM

(NON)UNIQUENESS OF MINIMIZERS IN THE LEAST GRADIENT PROBLEM WOJCIECH GÓRNY arxiv:1709.02185v1 [math.ap] 7 Sep 2017 Abstract. Minimizers in the least gradient problem with discontinuous boundary data need

### 1. Let A R be a nonempty set that is bounded from above, and let a be the least upper bound of A. Show that there exists a sequence {a n } n N

Applied Analysis prelim July 15, 216, with solutions Solve 4 of the problems 1-5 and 2 of the problems 6-8. We will only grade the first 4 problems attempted from1-5 and the first 2 attempted from problems

### Metric Space Topology (Spring 2016) Selected Homework Solutions. HW1 Q1.2. Suppose that d is a metric on a set X. Prove that the inequality d(x, y)

Metric Space Topology (Spring 2016) Selected Homework Solutions HW1 Q1.2. Suppose that d is a metric on a set X. Prove that the inequality d(x, y) d(z, w) d(x, z) + d(y, w) holds for all w, x, y, z X.

### BMO Round 2 Problem 3 Generalisation and Bounds

BMO 2007 2008 Round 2 Problem 3 Generalisation and Bounds Joseph Myers February 2008 1 Introduction Problem 3 (by Paul Jefferys) is: 3. Adrian has drawn a circle in the xy-plane whose radius is a positive

### Mathematics 2260H Geometry I: Euclidean geometry Trent University, Winter 2012 Quiz Solutions

Mathematics 2260H Geometry I: Euclidean geometry Trent University, Winter 2012 Quiz Solutions Quiz #1. Tuesday, 17 January, 2012. [10 minutes] 1. Given a line segment AB, use (some of) Postulates I V,

### J. HORNE, JR. A LOCALLY COMPACT CONNECTED GROUP ACTING ON THE PLANE HAS A CLOSED ORBIT

A LOCALLY COMPACT CONNECTED GROUP ACTING ON THE PLANE HAS A. CLOSED ORBIT BY J. HORNE, JR. The theorem of the title has its origin in a question concerning topological semigroups: Suppose S is a topological

### THEOREMS, ETC., FOR MATH 515

THEOREMS, ETC., FOR MATH 515 Proposition 1 (=comment on page 17). If A is an algebra, then any finite union or finite intersection of sets in A is also in A. Proposition 2 (=Proposition 1.1). For every

### USAC Colloquium. Bending Polyhedra. Andrejs Treibergs. September 4, Figure 1: A Rigid Polyhedron. University of Utah

USAC Colloquium Bending Polyhedra Andrejs Treibergs University of Utah September 4, 2013 Figure 1: A Rigid Polyhedron. 2. USAC Lecture: Bending Polyhedra The URL for these Beamer Slides: BendingPolyhedra

### Math 117: Topology of the Real Numbers

Math 117: Topology of the Real Numbers John Douglas Moore November 10, 2008 The goal of these notes is to highlight the most important topics presented in Chapter 3 of the text [1] and to provide a few

### Technical Results on Regular Preferences and Demand

Division of the Humanities and Social Sciences Technical Results on Regular Preferences and Demand KC Border Revised Fall 2011; Winter 2017 Preferences For the purposes of this note, a preference relation

### Foundations of Neutral Geometry

C H A P T E R 12 Foundations of Neutral Geometry The play is independent of the pages on which it is printed, and pure geometries are independent of lecture rooms, or of any other detail of the physical

### REVIEW OF ESSENTIAL MATH 346 TOPICS

REVIEW OF ESSENTIAL MATH 346 TOPICS 1. AXIOMATIC STRUCTURE OF R Doğan Çömez The real number system is a complete ordered field, i.e., it is a set R which is endowed with addition and multiplication operations

### arxiv: v1 [math.mg] 26 Jun 2014

From Funk to Hilbert Geometry Athanase Papadopoulos and Marc Troyanov arxiv:14066983v1 [mathmg] 26 Jun 2014 Institut de Recherche Mathématique Avancée, Université de Strasbourg and CNRS, 7 rue René Descartes,

### Strongly chordal and chordal bipartite graphs are sandwich monotone

Strongly chordal and chordal bipartite graphs are sandwich monotone Pinar Heggernes Federico Mancini Charis Papadopoulos R. Sritharan Abstract A graph class is sandwich monotone if, for every pair of its

### are Banach algebras. f(x)g(x) max Example 7.4. Similarly, A = L and A = l with the pointwise multiplication

7. Banach algebras Definition 7.1. A is called a Banach algebra (with unit) if: (1) A is a Banach space; (2) There is a multiplication A A A that has the following properties: (xy)z = x(yz), (x + y)z =

### Review for Grade 9 Math Exam - Unit 8 - Circle Geometry

Name: Review for Grade 9 Math Exam - Unit 8 - ircle Geometry Date: Multiple hoice Identify the choice that best completes the statement or answers the question. 1. is the centre of this circle and point

### GRE Math Subject Test #5 Solutions.

GRE Math Subject Test #5 Solutions. 1. E (Calculus) Apply L Hôpital s Rule two times: cos(3x) 1 3 sin(3x) 9 cos(3x) lim x 0 = lim x 2 x 0 = lim 2x x 0 = 9. 2 2 2. C (Geometry) Note that a line segment

### arxiv: v2 [math.mg] 24 Oct 2017

Farthest points on most Alexandrov surfaces arxiv:1412.1465v2 [math.mg] 24 Oct 2017 Joël Rouyer October 25, 2017 Abstract Costin Vîlcu We study global maxima of distance functions on most Alexandrov surfaces

### UTILITY OPTIMIZATION IN A FINITE SCENARIO SETTING

UTILITY OPTIMIZATION IN A FINITE SCENARIO SETTING J. TEICHMANN Abstract. We introduce the main concepts of duality theory for utility optimization in a setting of finitely many economic scenarios. 1. Utility

### = Find the value of n.

nswers: (0- HKM Heat Events) reated by: Mr. Francis Hung Last updated: pril 0 09 099 00 - Individual 9 0 0900 - Group 0 0 9 0 0 Individual Events I How many pairs of distinct integers between and 0 inclusively

### On Polya's Orchard Problem

Rose-Hulman Undergraduate Mathematics Journal Volume 7 Issue 2 Article 9 On Polya's Orchard Problem Alexandru Hening International University Bremen, Germany, a.hening@iu-bremen.de Michael Kelly Oklahoma

### Normal Fans of Polyhedral Convex Sets

Set-Valued Analysis manuscript No. (will be inserted by the editor) Normal Fans of Polyhedral Convex Sets Structures and Connections Shu Lu Stephen M. Robinson Received: date / Accepted: date Dedicated

### ON SETS OF CONSTANT WIDTH

92 GERALD FREILICH [February 2. A. E. Mayer, Grösste Polygone mit gegebenen Seitenvektoren, Comment. Math. Helv. vol. 10 (1938) pp. 288-301. 3. H. Hadwiger, Über eine Mittelwertformel für Richtungsfunktionale

### arxiv:math/ v1 [math.ho] 2 Feb 2005

arxiv:math/0502049v [math.ho] 2 Feb 2005 Looking through newly to the amazing irrationals Pratip Chakraborty University of Kaiserslautern Kaiserslautern, DE 67663 chakrabo@mathematik.uni-kl.de 4/2/2004.

### An introduction to some aspects of functional analysis

An introduction to some aspects of functional analysis Stephen Semmes Rice University Abstract These informal notes deal with some very basic objects in functional analysis, including norms and seminorms

### Convex Sets. Prof. Dan A. Simovici UMB

Convex Sets Prof. Dan A. Simovici UMB 1 / 57 Outline 1 Closures, Interiors, Borders of Sets in R n 2 Segments and Convex Sets 3 Properties of the Class of Convex Sets 4 Closure and Interior Points of Convex

### Sequences. Chapter 3. n + 1 3n + 2 sin n n. 3. lim (ln(n + 1) ln n) 1. lim. 2. lim. 4. lim (1 + n)1/n. Answers: 1. 1/3; 2. 0; 3. 0; 4. 1.

Chapter 3 Sequences Both the main elements of calculus (differentiation and integration) require the notion of a limit. Sequences will play a central role when we work with limits. Definition 3.. A Sequence

### Stanford Mathematics Department Math 205A Lecture Supplement #4 Borel Regular & Radon Measures

2 1 Borel Regular Measures We now state and prove an important regularity property of Borel regular outer measures: Stanford Mathematics Department Math 205A Lecture Supplement #4 Borel Regular & Radon

### McGill University Math 354: Honors Analysis 3

Practice problems McGill University Math 354: Honors Analysis 3 not for credit Problem 1. Determine whether the family of F = {f n } functions f n (x) = x n is uniformly equicontinuous. 1st Solution: The

### Sanjay Mishra. Topology. Dr. Sanjay Mishra. A Profound Subtitle

Topology A Profound Subtitle Dr. Copyright c 2017 Contents I General Topology 1 Topological Spaces............................................ 7 1.1 Introduction 7 1.2 Topological Space 7 1.2.1 Topological

### ON RIGIDITY OF ALGEBRAIC DYNAMICAL SYSTEMS

ON RIGIDITY OF ALGEBRAIC DYNAMICAL SYSTEMS ALEX CLARK AND ROBBERT FOKKINK Abstract. We study topological rigidity of algebraic dynamical systems. In the first part of this paper we give an algebraic condition

### Fuchsian groups. 2.1 Definitions and discreteness

2 Fuchsian groups In the previous chapter we introduced and studied the elements of Mob(H), which are the real Moebius transformations. In this chapter we focus the attention of special subgroups of this

### Test Corrections for Unit 1 Test

MUST READ DIRECTIONS: Read the directions located on www.koltymath.weebly.com to understand how to properly do test corrections. Ask for clarification from your teacher if there are parts that you are

### Near convexity, metric convexity, and convexity

Near convexity, metric convexity, and convexity Fred Richman Florida Atlantic University Boca Raton, FL 33431 28 February 2005 Abstract It is shown that a subset of a uniformly convex normed space is nearly

### Convex Sets with Applications to Economics

Convex Sets with Applications to Economics Debasis Mishra March 10, 2010 1 Convex Sets A set C R n is called convex if for all x, y C, we have λx+(1 λ)y C for all λ [0, 1]. The definition says that for

### ON THE NUMERICAL RANGE OF AN OPERATOR

ON THE NUMERICAL RANGE OF AN OPERATOR CHING-HWA MENG The numerical range of an operator P in a Hubert space is defined as the set of all the complex numbers (Tx, x), where x is a unit vector in the space.

### Introduction to Real Analysis

Introduction to Real Analysis Joshua Wilde, revised by Isabel Tecu, Takeshi Suzuki and María José Boccardi August 13, 2013 1 Sets Sets are the basic objects of mathematics. In fact, they are so basic that

### Math 9 Unit 8: Circle Geometry Pre-Exam Practice

Math 9 Unit 8: Circle Geometry Pre-Exam Practice Name: 1. A Ruppell s Griffon Vulture holds the record for the bird with the highest documented flight altitude. It was spotted at a height of about 11 km

### Examples of Dual Spaces from Measure Theory

Chapter 9 Examples of Dual Spaces from Measure Theory We have seen that L (, A, µ) is a Banach space for any measure space (, A, µ). We will extend that concept in the following section to identify an

### MORE ON CONTINUOUS FUNCTIONS AND SETS

Chapter 6 MORE ON CONTINUOUS FUNCTIONS AND SETS This chapter can be considered enrichment material containing also several more advanced topics and may be skipped in its entirety. You can proceed directly

### Problem set 1, Real Analysis I, Spring, 2015.

Problem set 1, Real Analysis I, Spring, 015. (1) Let f n : D R be a sequence of functions with domain D R n. Recall that f n f uniformly if and only if for all ɛ > 0, there is an N = N(ɛ) so that if n

### PROPERTIES OF C 1 -SMOOTH FUNCTIONS WITH CONSTRAINTS ON THE GRADIENT RANGE Mikhail V. Korobkov Russia, Novosibirsk, Sobolev Institute of Mathematics

PROPERTIES OF C 1 -SMOOTH FUNCTIONS WITH CONSTRAINTS ON THE GRADIENT RANGE Mikhail V. Korobkov Russia, Novosibirsk, Sobolev Institute of Mathematics e-mail: korob@math.nsc.ru Using Gromov method of convex

### Spring -07 TOPOLOGY III. Conventions

Spring -07 TOPOLOGY III Conventions In the following, a space means a topological space (unless specified otherwise). We usually denote a space by a symbol like X instead of writing, say, (X, τ), and we

### ANALYTIC TRANSFORMATIONS OF EVERYWHERE DENSE POINT SETS*

ANALYTIC TRANSFORMATIONS OF EVERYWHERE DENSE POINT SETS* BY PHILIP FRANKLIN I. Transformations of point sets There is a well known theorem in the theory of point sets, due to Cantor,t to the effect that