Invariant measure and geometric probability

Transcription

1 Proceedings of The Twelfth International Workshop on Diff. Geom. 12(2008) Invariant measure and geometric probability Jiazu Zhou, Min Chang and Fei Cheng School of Mathematics and Statistics, Southwest University, Chongqing, , People s Republic of China zhoujz@swu.edu.cn (2000 Mathematics Subject Classification : 52A22, 60D05.) Abstract. We begin with the measure for lines in plane and seek the connections between integral geometry and probability. Sylvester s theorem is obtained without knowing the measure on the set of lines in space. We also investigate the intrinsic geometric measures. 1 Introduction Integral geometry, referred to as geometric probability in the past and originated with the Buffon needle problem in 1770s, has classical roots in the work of Barbier, Crofton, Sylvester, Poincaré, and others. Later Blaschké, Chern, Weil, Santaló and others make contributions to this beautiful and fascinating branch of Mathematics (see [3], [5], [6], [7]). It begins with the study of invariant measures on geometric objects, such as the set of lines in the Euclidean plane and the set of linear spaces in the Euclidean space R n, and leads to integral identities and inequalities that have remarkable applications to probability and other mathematical branches. In the second half of last century another, related subject emerged in the work of I. M. Gelfand and S. Helgason and it is also called integral geometry. In contrast to its more classical counterpart, this newer type of integral geometry replaces geometric objects by functions and focuses on operations and convexity like the Radon transform. Geometric inequalities is one of the central branches of global differential geometry and geometric analysis and has been the object of much research in the last century and is still going on today. In general, geometric probability is the study of invariant measures. The probabilistic idea can be applied to obtain the containment measures. These containment measures could lead to geometric inequalities. Refer to [1], Key words and phrases: Geometric probability, geometric measure, invariant measure, Grassmanian. *) Supported in part by Chinese NSF (grant number: ) and Southwest University. 21

2 22 Jiazu Zhou, Min Chang and Fei Cheng [2], [8], [9], [10] for more details. A subset K in the Euclidean space R n is convex if any two points x, y of K are endpoints of the line segment lying inside K. We begin with the definition of an invariant measure and density of the lines in the Euclidean plane R 2 that will lead to calculate the measure of the set of lines that intersect with a convex set K. Then we will discuss the measure of the set of lines that intersect with a convex body in space. As an example, we will give Sylvester s theorem. More geometric probabilistic ideas on convex sets will be discussed. 2 Measure for sets of lines in plane R 2 A measure µ which takes real values and is not necessarily positive, is a function defined on a family of subset S that is closed under unions and intersections and contains the empty set [4]. Definition 1. Let F be a family of subsets of a set S in the n-dimensional Euclidean space R n, which is closed under unions and intersections, and contains the empty set. An invariant measure µ is a real valued function (not necessarily positive) defined on F and satisfies, 1. µ( ) = 0; 2. If A, B F are two measurable sets, then µ(a B) = µ(a) + µ(b) µ(a B); 3. For the group of rigid motions ISO(R n ) in R n, µ(a) = µ(ga); for g ISO(R n ). Note that a measure is characterized by the first two simple axioms. The third axiom means that the measure µ(a) of subset A is invariant under the rigid motions. That is, µ(a) is independent of the position of A in R n. Let X be a set of lines G(p, φ) : x cos φ + y sin φ p = 0; 0 p <, 0 φ < 2π.

3 Invariant measure and geometric probability 23 A motion g in R 2 can be expressed by g(a, b; α) : { x = x cos α y sin α + a, y = x sin α + y cos α + b, < a, b < + ; 0 α < 2π, where a, b, α are called the parameters of g and a, b are components of the translation, and α is the angle of rotation. We define the measure m(x) on X by the integral m(x) = f(p, φ) dp dφ, X where dg = f(p, φ)dp dφ is the invariant density and the function f is to be chosen such that the measure m(x) is invariant under the group of motions ISO(R 2 ) in the plane R 2. Under the motion g(a, b, α) the line G becomes gg : x cos(φ + α) +y sin(φ + α) a cos(φ + α) b sin(φ + α) p = 0, i.e., { p = a cos(φ + α) + b sin(φ + α) + p, φ = φ + α. It follows that dp dφ = dp dφ. Let X = gx, we have m(gx) = f(p, φ ) dp dφ X = f(a cos(φ + α) + b sin(φ + α) + p, φ + α)dp dφ. X The measure defined must satisfies the definition of a geometric measure 3). Therefore we have f(p, φ) = constant (almost everywhere). Let the constant be 1, then the measure of a set X of lines G(p, φ) is defined by the integral of the invariant density over the set X, i.e., m(x) = dg = dp dφ X dg = X dp dφ. It is an invariant measure under the group of motions in the plane and is unique up to a constant. The following known results are easy consequences (cf. [6], [7]).

4 24 Jiazu Zhou, Min Chang and Fei Cheng 1. Let L be a segment of length l in R 2. To get the measure of the set of lines that meet L. We take the origin of R 2 coordinates coincident with the middle point of L and the x-axis coincident with the direction of L. Then we have 2π m{g : G L } = dg = l 2 cos φ dφ = 2l. {G: G L } 0 2. Instead of L we consider a polygonal Γ of length l, composed of a finite number of segments L i of length l i, i.e., Γ = i L i and l = i l i. Let n = {G Γ}, the number of points that G intersects Γ, then we have n dg = 2l. ( ) {G Γ } Proposition 1. (Crofton [6]) Let K be a convex set in R 2. Then we have dg = dg = Length( K). {G K } {G K } Actually by a limit process it is not difficult to prove that (*) holds for any rectifiable curve. Proposition 2. (Blaschké [6]) Let Γ be a simple closed rectifiable curve of length l in R 2. Let n = {G Γ}, the number of points that G intersects Γ, then we have {G Γ } n dg = 2l. 3 Buffon Needle problem and the estimation for π Assume that plane R 2 is covered with the net N(d), that is, parallel lines at a distance d from each other. A needle L of length l is dropped at random on the plane. What is the probability that the needle shall meet the net (i.e., at least one of the lines). This problem can be solved by many ways, for example, by computation with conditional probability [3]. It is, however, more instructive to solve it by another

5 Invariant measure and geometric probability 25 method (Klain and Rota), one that minimizes the computation and maximizes the role of probabilistic idea. Let X 1 be the number of intersection of a randomly dropped needle of length l 1 with any of the parallel straight lines. If the needle is short, say l 1 < d, then X 1 can takes only 0 or 1. Let p n denote the probability that the needle meets exactly n straight lines and E(X 1 ) denote the expectation of the random variable X 1, then we have E(X 1 ) = n 0 np n. Thus if l 1 < d, then E(X 1 ) = 0p 0 + 1p 1 = p 1, and p 1 is the probability we seek. Therefore, it is sufficient to compute the expectation E(X 1 ). Suppose that another needle of length l 2 is dropped at random. The number of intersections of the second needle with any of the parallel lines on R 2 is another random variable X 2. The random variables X 1 and X 2 are independent unless the needles are welded together at one of their endpoints. They may form a straight line or they may be an angle if two needles are welded together at one of their endpoints. In any cases, if the two rigidly bound needles are simultaneously dropped on R 2, their total number of intersections will be X 1 + X 2. These two random variables X 1 and X 2 will no longer be independent, but their expectation will remain additive: (3.1) E(X 1 + X 2 ) = E(X 1 ) + E(X 2 ). The same reasoning applies to the random variable X 1 + X X k, in the case that k needles are welded together to form a polygonal lines of arbitrary shape. Since E(X 1 ) clearly depends on the length l 1, we can write E(X 1 ) = f(l 1 ), where f is a function to be determined. By welding together two needles so that they form one straight line we find that E(X 1 + X 2 ) = f(l 1 + l 2 ), and we have (3.2) f(l 1 + l 2 ) = f(l 1 ) + f(l 2 ). It then follows that f is linear when restricted to rational values of l. Since f is clearly a monotonically increasing function with respect to l, we infer that f(l) = rl for all l R by Cauchy integral recursion theorem, where the constant r is to be determined. If Γ is a rigid wire of length l, dropped randomly on R 2, and if Y is the number of intersections of Γ with any of the straight lines, then Γ can be approximated by polygonal wires, so that Y is approximately equal to X X k. Passing to the limit, we find that (3.3) E(Y ) = rl. Then we can determine r by choosing a wire of suitable shape. Let Γ be a circular wire of diameter d. Obviously E(Y ) = 2 except for a set of measure zero, and l = πd. Then we have 2 = rπd,

6 26 Jiazu Zhou, Min Chang and Fei Cheng and thus we have r = 2 πd, for a short needle, and the Buffon s result (3.4) E(X 1 ) = p 1 = 2l πd. This result had been used to compute the value of π in the past. If we drop the needle N times and count the number n that the needle hits the net. Here is a historic data for d = 1 [5]: l N n value of π Wolf, Smith, De Morgan, Fox, Lazzerini, Reina, Gridgeman, We shall use this as the idea leading into the heart of geometric probability. 4 Sylvester s theorem Let G (n,k) denote the set of all linear varieties of dimension k in space R n, not necessarily through the origin, that is called the Grassmanian. The group ISO(R n ) of Euclidean rigid motions acts on the Grassmanian, and there is an invariant measure on the Grassmanian under the action of ISO(R n ). This invariant measure is unique except for a constant factor. Computation with invariant measures on G (n,k) is complicated (see [5, 6]) and we will not going to pursue the explicit formula for the invariant measure on G (n,k) here. We denote λ n k the invariant measure on G (n,k). Let G (2,1) denote the set of all straight lines in plane R 2. Denoted by X the number of intersections of a random straight line G with a straight segment L of length l, and let λ 2 1 denote the invariant measure on G (2,1), then the integral G (2,1) X dλ 2 1 depends only on l and it is equal to the measure of the set of lines that meet with the given segment L, that is G (2,1) X dλ 2 1 = m{g : G L } = f(l)

7 Invariant measure and geometric probability 27 since X takes only 0 or 1. Now we repeat the same process we used in the Buffon needle problem. Let Γ be a polygon consisting of segments L k (k = 1,, m) of length l k and X k the number of intersections of a random line with segment L k. Then we have G (2,1) (X X m ) dλ 2 1 = f(l l m ). Since the integral is linear and we have X 1 dλ X m dλ 2 1 = f(l 1 ) + + f(l m ). G (2,1) G (2,1) Also via Cauchy integral recursion theorem we have f(l) = rl. One can normalize the measure λ 2 1 by letting r = 1. Let K be a convex set and let X K be the number of intersections of K with a random line. Then G (2,1) X K dλ 2 1 = r Length( K). In particular, let K k (k = 1, 2) be two convex sets in the plane with non-empty interiors. For each k, we have X Kk dλ 2 1 = r Length( K k ). G (2,1) Since a straight line meets with a convex set either twice or not at all except for a measure 0 set of lines, therefore we have X Kk dλ 2 1 = 2 dλ 2 1, G (2,1) G (2,1) the twice of the measure m{g : G K k }. The following known theorem is an application of our results to probability. Theorem 1. (Sylvester) Let K 1 and K 2 be two convex sets in plane R 2 and suppose that K 1 is contained in K 2. Then the conditional probability that a random straight line shall meet K 1, given that it meets K 2, is the ratio p{g K 1 G K 2 } = Length( K 1) Length( K 2 ). Note that the Sylvester s theorem may be restated as its dual form:

8 28 Jiazu Zhou, Min Chang and Fei Cheng Theorem 2. Let K 1 and K 2 be two convex sets in plane R 2 and suppose that K 1 is contained in K 2. Then the conditional probability that a random point P shall belong to K 1, given that it belongs to K 2, is the ratio p{p K 1 P K 2 } = Area(K 1) Area(K 2 ). Let us consider G (3,1) (which could be viewed as the 2 dimensional projective plane if all straight lines passing through the origin) and λ 3 1, the invariant measure on the set of all straight lines in three dimensional Euclidean space R 3. Consider a rectangle R replaces in R 3, and consider the set of all straight lines that meet R. We wish to compute the measure of this set of lines without knowing the formula of λ 3 1. Since a straight line meets the rectangle R at a point or not at all, therefore the value of the measure of the set of all lines meeting with R depends only on the Area(R) of the rectangle R. If we take another rectangle R whose areas is double the area of R, then the measure of the sets of all lines meeting R is double the measure of the set of all lines meeting R. Proceeding along these lines, we get to Cauchy integral equation, and we infer that the measure of the set of all straight lines meeting a rectangle R equals a constant times the Area(R). We may normalize the measure λ 3 1 and set this constant equal to one. We could have worked with any planar figure Ω whatsoever, placed in an arbitrary position in space. The measure of set of lines meeting Ω equals the Area(Ω), by the same reasoning and the limit process. Let us take a set F in R 3 which is the union of disjoint sets Ω 1, Ω 2,, Ω k, where each Ω i is contained in a different plane. We wish to find the measure of the set of all straight lines meeting F without knowing the formula for the invariant measure λ 3 1. Such a computation can be carried out theoretically but it is very complicated. We change the problem so the probabilistic idea could work. Instead of computing a measure, let us compute the integral X F (ω)dλ 3 1(ω), G (3,1) where X F (ω) is the number of times the straight line ω meets F and ω ranges over the Grassmanian Gr (3,1), the set of all straight lines in space R 3. We can compute this integral without knowing the measure λ 3 1 on Gr (3,1). Since and since the Ω i are disjoint, we have F = G (3,1) X F (ω)dλ 3 1(ω) = k Ω i, i=1 k i=1 Gr (3,1) X Ωi (ω)dλ 3 1(ω).

9 Invariant measure and geometric probability 29 Since we have chosen each of the sets Ω i to lie in a plane, so that a straight line meets Ω i either once or not at all. It follows that G (3,1) X Ωi (ω)dλ 3 1(ω) = Area(Ω i ) and hence G (3,1) X F (ω)dλ 3 1(ω) = k Area(Ω i ) = Area(F ). We can pass to the limit and obtain the following assertion. i=1 Theorem 3. Let S be any measurable surface in space, and let X S (ω) be the number of times the straight line ω meets the surface S. Then we have G (3,1) X S (ω)dλ 3 1(ω) = Area(S), where Area(S) is the surface area of S. In probabilistic language: that the average number of times a randomly chosen straight line meets the surface S equals the surface area of S. For a convex set K we have, let X K (ω) be the number of times the random straight line ω meets K. Then X K (ω) equal 2 or 0 except for a measure zero set of lines, Then we have X K (ω)dλ 3 1(ω) = 2 dλ 3 1(ω) = Area(K). {ω K } {ω K } A plane rectangle R may be viewed as a convex set of area 2Area(R) and hen we have dλ 3 1(ω) = Area(R). {ω G (3,1) ω R } Let us repeat the same processing by take the set of all planes in space, instead of the set of all straight lines. The invariant measure on this Grassmanian G (3,2) is denoted by λ 3 2. Since a plane meets a straight line segment either in a point or not at all, the same argument shows that the measure of sets of all planes that meet a line segment L equals Length(L), the length of the segment L. More generally, repeating the process we have Let Γ be any rectifiable curve in space R 3, and if X Γ (π) equals the number of times the plane π meets the curve Γ, then we have G (3,2) X Γ (π)dλ 3 2(π) = Length(Γ). Let us back to the probabilistic problem.

10 30 Jiazu Zhou, Min Chang and Fei Cheng Let K 1 and K 2 be two compact convex sets in R 3 and suppose that K 1 is contained in K 2. Assuming that a random straight line meets the large set K 2, what is the probability that such a straight line will also meet the the smaller set K 1? This leads to Theorem 4. (Sylvester) Let K 1 and K 2 be two convex sets in R 3 and suppose that K 1 is contained in K 2. Then the conditional probability that a random straight line shall meet K 1, given that it meets K 2, is the ratio p{g K 1 G K 2 } = Area(K 1) Area(K 2 ). The dual of above Sylvester theorem can be restated as. Theorem 5. Let K 1 and K 2 be two convex sets in R 3 and suppose that K 1 is contained in K 2. Then the conditional probability that a random point shall belong to K 1, given that it belongs to K 2, is the ratio p{p K 1 P K 2 } = Volume(K 1) Volume(K 2 ). The similar discussion can be carried on G ( n, k). coming papers. We will investigate in the Acknowledgement. The first author would like to thank Professor Young Jin Suh for his invitation to the International Workshop on Integral Geometry, Submanifolds and Related Topics, which is held in Kyungpook National University during May 25-26, Most part of this paper comes from the lectures at Taegu in Korea. References [1] E. Grinberg, D. Ren, & J. Zhou, The symmetric isoperimetric deficit and the containment problem in a plan of constant curvature, preprint. [2] E. Grinberg, S. Li, G. Zhang & J. Zhou, Integral Geometry and Convexity, World Scientific, Singapore (2006). [3] D. Klain & G-C. Rota, Introduction to Geometric Probability, Cambridge University Press (1997). [4] S. Lang, Real Analysis, Addison-Wesley Publishing Company (1983). [5] D. Ren, Topics in Integral Geometry, World Scientific, Sigapore (1994). [6] L. A. Santaló, Integral Geometry and Geometric Probability. Addison-Wesley, Reading, Mass. (1976).

11 Invariant measure and geometric probability 31 [7] R. Schneider, Convex Bodies: The Brunn-Minkowski Theory, Cambridge University Press (1993). [8] G. Zhang & J. Zhou, Containment measures in integral geometry, Integral geometry and Convexity, Editored by E. Gringberg, S. Li, G. Zhang & J. Zhou, World Scientific, Singapore (2006), [9] J. Zhou & F. Chen, The Bonnesen-type inequalities in a plane of constant curvature, Journal of Korean Math. Soc., Vol.44 (2007), No.6, [10] J. Zhou, On plane Bonnesen-type inequalities, Acta Math. Sinica, Vol.50, No.6 (2007),