On a formula for sets of constant width in 2d

Transcription

1 On a formula for sets of constant width in 2d Bernd Kawohl, Guido Sweers May 28, 28 Abstract A formula for smooth orbiforms originating from Euler can be adjusted to describe all sets of constant width in 2d. Moreover, the formula allows short proofs of some laborious approximation results for sets of constant width. Introduction Around 774 Leonhard Euler considered curves, which he called curva orbiformis. A formula describing such curves can be found in of [9]. In a series of papers [], [], [2] in the 95 s Hammer and Sobczyk recalled the formula to construct all curves of constant breadth in the plane. They started from a characterization of outwardly simple line families F = ϕ F ϕ, with { F ϕ = y R 2 ; y (cos } ϕ sin ϕ = q (ϕ for all ( sin ϕ S. Since the orientation of a line should play no role, the Lipschitz-continuous function q satisfies q (ϕ + = q (ϕ for all ϕ. ( They use q to define a curve x : [, 2] R 2 of constant breadth by ( (cos x (ϕ = c q (s ds ϕ ( q (ϕ sin ϕ. (2 Indeed, for c large enough in (2, that is, c sup ϕ { Dq (ϕ + sin ϕ } q (s ds, (3 q(ϕ+h q(ϕ with Dq (ϕ = lim sup h h, the function x describes a curve of constant breadth and, moreover, each curve of constant breadth can be obtained this way. The detour through outwardly simple line families and the result spread over three papers does not make it simple for the reader. Over the years mathematicians have been intrigued by sets of constant width and we would like to mention some of the papers of a more general nature: [8, 9], [3], [7], [8], [6] and [5]. Also several books dealt with the subject [4], [3], [6]. Relatively elementary sets of constant width in two dimensions that can be constructed geometricallly, are the so called Reuleaux polygons. These sets are defined by having boundaries consisting of finitely many circular arcs. See the two examples in Fig.. The obvious question is, whether one can approximate any set of constant width by a set of constant width that is a Reuleaux polygon. In 95 Blaschke [4] proved such a result in two dimensions. In 95 Jaglom and Boltjanski published in Russian the result that all arcs in the approximating Reuleaux polygon can be choosen to have the same curvature. Translations appeared in 956 in German [4] and in 96 in English. Their proof is Universität zu Köln, Mathematisches Institut, Weyertal 86-9, 593 Köln, Deutschland.

2 purely geometrical and fills several pages. Also Meissner, who was interested in the applications of sets of constant width, focused on geometry in [8, 9] around 99. The approximations of Tanno in [2] and Wegner in [2] went in the other direction, showing that any set of constant width in two dimensions can be approximated by such sets having a smooth and even analytical boundary. Figure : Two sets of constant width with a boundary of circular arcs. On the left the Reuleaux triangle. The segments connect boundary points with opposite normal directions for which the constant width is attained. The points denote the centers of rotation. In 2 dimensions there are both the analytical approach originating from Euler as well as the geometrical approach; for example from Meissner. We want to unify these approaches and we will do so by modifying the formula in (2 to one that displays some geometrical quantities. The modified formula is not entirely new. The rudimentary idea seems to be given by Barbier [2] in 86, and Blaschke [4] sort of mentions it in footnote about an alternative way to describe the area-minimisation problem for sets of constant width, but he does not give a derivation or reference. In Section 3 we will recall the modified formula. Section 4 is devoted to the converse, namely that any set of constant width is described by this formula. Moreover, the formula will be more convenient to give short proofs of the approximations we have just mentioned. This can be found in Section 5. Let us start by recalling what is today called a set of constant width in R n. Definition A closed, bounded and convex set G R n L R + if the following holds for each x G: is called a set of constant width. for all y G one has x y L; 2. there exists y G with x y = L. An equivalent way of defining a set of constant width G is to say that the width of G in direction ω S n, given by d G (ω = max {x ω; x G} min {x ω; x G}, does not depend on ω. Here S n = {ω R n with ω = }. Again this can be reformulated by defining for a closed bounded set G the support function p G : S n R by p G (ω = max {x ω; x G}. (4 Note that d G (ω = p G (ω + p G ( ω. Meissner [8] already knew that a closed, bounded and convex set G is a set of constant width L if and only if p G (ω + p G ( ω = L. (5 2

3 d G (ω ω ω x ω G x ω Figure 2: The relation between the directional width and the support function Since being convex and of constant width implies being strictly convex, a set of constant width G has for each ω S n a unique x ω G such that p G (ω = x ω ω (6 and ω is an outer normal direction at x ω. Note that x G maybe reached by several ω S n : the set { ω S n ; x = x ω } is connected, but not necessarily single valued. 2 The construction by Euler This corresponding formula, recalling (2, ( x (ϕ = c q (s ds indeed yields the following result: (cos ϕ sin ϕ ( q (ϕ sin ϕ, Theorem 2 (Euler, Hammer-Sobczyk For any function q C, (R satisfying ( and for any constant c satisfying (3 the function x defined in (2 describes the boundary of a set of constant width in R 2. Proof. Before giving the proof we should point out that the width 2r of this set can be expressed in terms of q by (5 and (8 below. Notice that ( implies that q is 2-periodic and that +2 ϕ q (s ds = for all ϕ. So x as in (2 is 2-periodic and moreover, for q C (R one finds that ( ( x (ϕ = c q (ϕ q (s ds sin ϕ. (7 For q C, (R we want to use an approximation argument that preserves the conditions in ( and (3. To do so we consider the function g L (R, defined by g (ϕ = q (ϕ + q (s ds, (8 which is differentiable almost everywhere The function q we recover from q ( and g through q (ϕ = q ( + 3 cos (ϕ s g (s ds. (9

4 Note that the 2-periodicity of g shows the 2-periodicity of q. Moreover, the formula in (9 gives a solution to the integro-differential equation (8. Indeed, and q (ϕ = q ( sin (ϕ q (s ds = q ( sin (ϕ + = q ( sin (ϕ + = q ( sin (ϕ + s sin (ϕ s g (s ds + g (ϕ t cos (t s g (s dsdt cos (t s dt g (s ds sin (ϕ s g (s ds. ( Using a mollifier to approximate g by g n we find g n g L p for all p and also g n g. Through (9 one finds from g n g L p that q n q. For c satisfying (3 it follows from g n g that c q n (ϕ q n (s ds. ( Thus, the smooth function q n inherits the properties of q in ( and satisfies (3 with the same constant c as for q. Hence we may proceed by assuming that q is smooth. With the property in ( and the formula in (7 one finds that ϕ x (ϕ rotates counterclockwise. Moreover, the outside normal ν (ϕ at x (ϕ, when x (ϕ, satisfies ν (ϕ = ( sin ϕ. The curvature satisfies κ (ϕ = x (ϕ x 2 (ϕ x (ϕ x 2 (ϕ = (x (ϕ2 + x 2 (ϕ2 3/2 c q (ϕ q (s ds (, ]. Being 2-periodic, rotating to the left and κ > imply that x ([, 2] is the boundary of a well-defined strictly convex domain. Moreover, (2 implies with ( x (ϕ + x (ϕ = and, with p G as in (6, ( c + + (cos q (s ds c + q (s ds ϕ sin ϕ + + (q (ϕ + + q (ϕ ( ( sin ϕ = 2c + q (s ds (cos ϕ sin ϕ ( p cos(ϕ+ ( G sin(ϕ+ + ( ( pg sin ϕ = x (ϕ + + x (ϕ sin ϕ = 2c q (s ds. (2 A convex domain for which the expression (2 is constant is a set of constant width. 3 A modified construction We will not use (2 directly. Our related formulation was inspired by a video on the educational website [22]. After 3 minutes and seconds this video shows an informative recipe for constructing sets of constant width consisting of circular arcs by rotating sticks of length 2r over a variable center located on the stick with the center at a signed distance a (ϕ from its center. The video uses a piecewise constant function a. It is not mentioned how the values were chosen. But obviously one cannot take arbitrary values if the curve has to be closed. As mentioned before, the formula was known to Blaschke [3]. Even Barbier seems to explain the construction as early as 86, but only in words [2, Footnote on Page 279]. The video motivated us to ask a under what condition on a does this construction work, b if one can generalise the construction to functions a L, and c if all sets of constant width can be constructed in this way. The first two questions are treated in this section and the last one in the next section. 4

5 Recipe 3 For a L (R with and x R 2 set a (ϕ + = a (ϕ for all ϕ, (3 a (s ( ( sin s cos s ds =, (4 x (ϕ = x + r a (5 (r a (s ( sin s cos s ds. (6 Blaschke must have known the formulation since only then does the remark in [4, Footnote** on page 55f] showing conditions (4 and (5 make sense. He, however, gives no reference, nor does he seem to make any further use of it. For q smooth one may translate (2 into (6, with the assumptions in Recipe 3 satisfied, through x = ( a (ϕ = 2 c q(, r = c 2 q (s ds q (ϕ q (s ds and (7 q (s ds. (8 So one finds that g from (8 is related to a by a + g = 2 q (s ds. However, instead of transferring the formula in (6 back to (2 we prefer to give a direct proof that (6 describes the boundary of a set of constant width. Theorem 4 With a, r and x as in Recipe 3 the function x : [, 2] R 2 in (6 describes a closed curve through x and x ( 2r, which is the boundary of a set of constant width 2r. Examples relating a and x for r = a can be found in Fig. 3. Proof. Setting one finds u (ϕ := u (ϕ = (r a (s sin (ϕ s ds = r ( (r a (s cos (ϕ s ds = r sin ϕ a (s sin (ϕ s ds (9 a (s cos (ϕ s ds, and u (ϕ = r a (ϕ u (ϕ a.e. (2 Since a satisfies (3, one finds that u is periodic with period 2. With some straightforward calculations it follows that x from (6 can be rewritten to x (ϕ = x + u (ϕ ( sin ϕ + u (ϕ ( sin ϕ, (2 showing that x is periodic. The derivative of x is defined almost everywhere and then satisfies x (ϕ = (r a (ϕ ( sin ϕ and κ (ϕ = r a (ϕ. (22 Since r a, it follows from the sign of the integrand in (9 that u (ϕ for ϕ [, ] and hence on R and with (22 that x rotates counterclockwise. Since the curvature satisfies [ ] κ (ϕ r + a,, 5

6 Out[386]= Out[34]= - - Out[359]= Out[83]= - - Out[848]= Out[395]= - - Out[524]= Out[578]= - - Figure 3: Graphs of the functions a in Recipe 3 with the corresponding sets of constant width for r = a. the set x ([, 2] is the boundary of a well-defined strictly convex domain. From (2, (9 and (3, (4 we also find that p ( ( cos(ϕ+ sin(ϕ+ + p ( sin ϕ = ( x (ϕ + + x (ϕ sin ϕ = = u (ϕ + + u (ϕ = r ( cos (ϕ + + r ( = 2r. By (5 this implies that the enclosed domain is a set of constant width. The conditions u ( = u ( = imply x ( = x. Since u ( = 2r and u ( = imply x ( = x ( 2r, the claim follows. As mentioned above one can see the construction in (6 of Recipe 3 as the rotation of a stick of length 2r over a variable center located on the stick with the center of rotation at a distance a (ϕ from its center. The inequality in (5 takes care that this center of rotation indeed lies inside the stick. In fact this rotation center reaches r as the smallest distance to one of the ends when r = a. One also finds that the maximal curvature of the set of constant width is (r a for r > a and infinity for r = a. The minimal curvature is (r + a. The conditions in (3 and (4 make sure that ends meet. If one wraps a set G of constant width L in a layer of width ε, i.e. if G ε := {x + y ; x G, y B ε (} is the Minkowski sum of G with a ball of radius ε, then G ε has constant width L + 2ε. So one may modify r and mollify corners by adding any positive constant ε to find that r + ε also gives a set of constant width. 6

7 In the case that a is piecewise constant, one finds a set of constant width with a boundary consisting of finitely many circular arcs. This is nicely illustrated in [22]. 4 From the set of constant width to the function a In this section we will show the converse of the result in Theorem 4. Theorem 5 Let G R 2 be a set of constant width. Then there exist a, r and x as in Recipe 3 such that x ([, 2] = G. More precisely:. For x the point on G with maximal x -coordinate and x the point on G with minimal x -coordinate, one has r = 2 x x. 2. For κ min R + the( minimal curvature of G and κ (, ] the maximal curvature, one has a = 2 κ min ( κ max and r 2 κ min + max κ. Remark 5. The Reuleaux triangle in Fig. has r = and for t (, 3, a(t := for t ( 3, 2 3, for t ( 2 3,. One Proof. findsfix κ x min, = x 2 and rκ as stated max =. in the Proposition. Since G is a set of constant width, for each ω = ( sin ϕ S there is a unique point x ω G such that Proof. Fix x, x and r as stated in the theorem. Since G is a set of constant width, for each ω = ( sin ϕ S x ω < x there is a unique point ω ω for all x G \ {x x ω G such that ω }. Instead of G we consider G ε = { } B x ω ε (x; x G, which is again a set of constant width 2r+2ε. < ω ω for all x G \ {x ω }. Moreover, if x ε ω denotes the point Instead of G we consider G ε = with { } x ω < xb ε ε (x; x G, which is again a set of constant width 2r+2ε. ω ω for all x G ε \ {x ε ω}, Moreover, if x ε ω denotes the point with we find x ε ω = x ω + εω. Since G ε is convex, Alexandrov s Theorem implies that the curvature of G ε is defined almost everywhere. Moreover, x ω < x ε since each x ω ω for all x G ε ω G ε \ {x ε lies ε on the boundary of ω}, B ε (x ω and some B 2r+ε (x ω with we find x ε ω = x ω + εω. Since GB ε (xis ω convex, G ε BAlexandrov s 2r+ε (x ω Theorem implies that (22 the curvature of G ε is defined almost everywhere. Moreover, since each x] ε ω G ε lies on the boundary of B ε the (x ω curvature and some κ (x ε ω B 2r+ε of G(x ε at ω x ε ωwith lies in the interval [(2r + ε, ε. See Fig 4. B ε (x ω G ε B 2r+ε (x ω, (24 ] the curvature κ (x ε ω of G ε at x ε ω lies in the interval [(2r + ε, ε. See Fig 4. (23 Figure 4: G and G ε with the auxiliary circles Letting ω = ( sin ϕ S be some direction and x ε ω one may parameterise the boundary locally by be the corresponding boundary point, t X ε (t := x ε ω + t ( sin ϕ 7 ( y (t sin ϕ,

8 where y is a C, -function with y ( = y ( = and y defined almost everywhere. One finds for the outside normal and curvature: n (X ε (t = +y (t 2 ( ( sin ϕ + y (t ( sin ϕ κ (X ε (t = (+y (t 2 3/2 y (t Then the angle ϕ is parameterised by ϕ = Φ ε (t := ϕ + arcsin and Φ ε C, is differentiable almost everywhere with Φ ε (t = y (t +y (t 2 a.e.,, (25 [ (2r + ε, ε ] a.e. (26 ( y (t +y (t 2 which is strictly positive and bounded for t small by (26. So locally Φ inv ε exists, lies in C,, is differentiable almost everywhere and locally near ϕ with a strictly positive and bounded derivative where it is defined. By choosing several ϕ we may parameterise G ε by a function ϕ x ε (ϕ : [, 2] G ε that lies in C, and is diffferentiable almost everywhere. The construction of x ε is such that (x ε (ϕ x ε (ϕ + (cos ϕ sin ϕ = x ε (ϕ x ε (ϕ + = 2r + 2ε, (27 (x ε (ϕ x ε (ϕ + ( sin ϕ = and (28 x ε (ϕ ( sin ϕ By differentiating (28 and using (27 we find that and hence by (29 (x ε (ϕ x ε (ϕ + ( sin ϕ = x ε (ϕ a.e. (29 = 2r + 2ε a.e. x ε (ϕ + + x ε (ϕ = (x ε (ϕ x ε (ϕ + ( sin ϕ Since x ε (Φ ε (t = X ε (t, we find for ϕ = Φ ε (t that + y (t 2 Hence x ε (ϕ = X ε (t Φ ε (t = = 2r + 2ε a.e. = κ (x y (t ε (ϕ [ε, 2r + ε] a.e. +y (t 2 κ (x ε (ϕ + κ (x ε (ϕ + = 2r + 2ε a.e. The function a L (, 2 is then well-defined by: a (ϕ = 2 ( x ε (ϕ + x ε (ϕ = ( 2 κ (x ε (ϕ + κ (x ε (ϕ and satisfies a (2r + ε ε = r. 2 This definition implies directly that a (ϕ + = a (ϕ and, moreover, that r + ε a (ϕ = r + ε 2 ( x ε (ϕ + x ε (ϕ = r + ε 2 (2r + 2ε 2 x ε (ϕ = x ε (ϕ and hence (r + ε a (ϕ ( sin ϕ = x ε (ϕ ( sin ϕ = x ε (ϕ, 8

9 which in turn implies that Since x ε ( ( 2r+2ε x ε (ϕ = x ε ( + the conditions in (4 hold as well. s = xε ( = x ε ( + = x ε ( ( 2r+2ε (r + ε a (s ( sin s cos s ds. (r + ε a (s ( sin s cos s ds a (s ( sin s cos s ds, 5 Applications By the formula in (6 one may find rather straightforward approximation results for sets of constant width in 2 dimensions. The function a that determines the set is in L (, 2 and can be approximated a in the L 2 -topology by step functions and b in the weak L 2 topology by functions that oscillate finitely often between just two values ±r. Moreover, it can be approximated in L 2 c by functions of class C using convolution with Friedrichs mollifiers ζ ε, or d by analytic functions using partial sums of Fourier series. Any approximation of a in L 2 or weak L 2 implies an approximation of the underlying set defined by (6 in the Hausdorff-metric. In the following list of corollaries we approximate a along the lines of a through d, but we must also make sure that our approximations satisfy (3,(4 and (5. Blaschke proved in 95 [4] that two-dimensional sets of constant width 2r can be approximated in C by Reuleaux polygons of constant width 2r + 2ε, that is, by domains having a boundary consisting of circular arcs. Using (6 we can replace the geometric proof by an analytic one. Corollary 6 Let γ (, 2. Any two-dimensional set of constant width can be approximated in C,γ -sense by a Reuleaux-polygon of constant width. Remark 6. Let G and G 2 be two bounded domains, k N and γ [, ]. We say G and G 2 are ε-close in X-sense, when there exist parameterisations γ : M G and γ 2 : M G 2 with γ γ 2 X ε. Proof. Continuous functions are dense in L 2 (,. Choose a ε C [, ] such that a a ε L2 (, < ε, then, since a r, also a if a ε (x > a, a ε (x = a ε (x if a ε (x a, a if a ε (x < a, is continuous, satisfies a a ε L 2 (, < ε and additionally a ε r. The function a ε can be approximated uniformly by the step function a ε,n (x = 2n k= a ε ( k+/2 2n [ k k+ (x. 2n, 2n Choosing n large enough we find a a L2 ε,n < 2ε and also, for cs either cos or sin, that (, c cs = a ε,n (x cs (x dx 9

10 satisfies c cs = ( a ε,n (x a (x cs (x dx a a L ε,n 2 (, < 2ε. Then ã ε (x = a ε,n (x 2 (c sin + c cos [, (x 2 2 (c sin c cos [ 2, (x is a stepfunction with a ã ε L 2 (, < ( ε which satisfies (4. Moreover, ã ε a ε,n + 2ε a + 2ε. Thus, by changing r to r + 2ε also (5 is satisfied. With a approximated in L 2 -sense one finds that the curve defined in (6 is approximated in H -sense which, by Sobolev, is embedded in C,γ for γ < 2. The next result from 95 is stated and proved by Jaglom and Boltjanskii [4, p. 63 ff] using a geometrical approach. It claims that the approximation by Reulaux polygons can be restricted to those polygons whose circular arcs have identical curvature /r. Their lengthy geometric proof can be shortened using (6. Corollary 7 Every set of constant width d in 2 dimensions can be approximated in C -sense by sets of constant width d that have a boundary which consists of circular arcs with curvature /d. Before proving Corollary 7 we need an auxiliary lemma. Lemma 8 Let < β <. For each ρ L (, β with ρ, there exists an interval [γ, δ] [, β] such that ρ (x e ix dx = δ γ e ix dx. (3 e iβ (β j ( j ( e i (β Figure 5: The dark (red part contains the values ρ (s eis ds for all allowed ρ. The bounding curves j (, j ( correspond to the extreme cases in (33 as function of θ: j (θ = +θ(β e is ds and j (θ = β θ(β eis ds. Proof. Fix θ (, and consider all ρ as in Lemma 8 with average θ, that is, β ρ (x dx = θ. (3 β For β < one finds ( ( ( := arg ρ (x e ix dx arg ρ (x e ix dx arg ρ (x e ix dx =: β, (32

11 where ρ (x = { for x + θ (β, for x > + θ (β, and ρ (x = { for x β θ (β, for x > β θ (β. (33 Indeed Pontryagin s maximum principle [7], helps us to determine the minimum for ( J := Re e i(x ρ (x dx = cos (x ρ (x dx among all ρ satisfying (3. We take v (t = ρ (t with v ( = and v (β = θ (β, H (v (t, ρ (t, λ (t, t = λ (t ρ (t + cos (t ρ (t. Pontryagin states that for the optimal trajectory (v, ρ, λ that minimizes H with λ piecewise continous, it holds that λ (t ρ (t + cos (t ρ (t λ (t ρ (t + cos (t ρ (t for all ρ (t [, ]. So if ρ (t =, then λ (t + cos (t. If ρ (t > then we get a contradiction, unless ρ (t = and λ (t + cos (t. Hence ρ (t {, }. A similar result shows the right hand side of (32. Through approximations by stepfunctions and rearrangement one finds that the optimal cases are as in (32. For these cases, that is ρ and ρ as in (33, one has ρ (x e ix dx = +θ(β e ix dx and ρ (x e ix dx = β θ(β e ix dx. For all other ρ with ( β ρ (x dx = θ (β one finds that β ρ := arg ρ (x eix dx (, β. For the norm with those ρ one finds ρ (x e ix dx ρ (x e ix dx. lies in By rearrangement there is ρ [, ] such that ρ+ ρ (x e ix 2 θ(β dx = ρ e ix dx ρ 2 θ(β for some [ ρ 2 θ (β, ρ + 2 θ (β ] [, β]. If ρ =, we are done. If ρ <, there exists s > such that ρ ρ+ 2 θ(β ρ 2 θ(β e ix dx = Since θ (, was arbitrary, the claim follows. Proof of Corollary 7. By Lemma 8 there is [c k, d k ] [ k 2n So we define ρ+ 2 θ(β s ρ 2 θ(β +s e ix dx. We divide the interval [, ] in 2n subintervals {[ k 2n k+ 2n k 2n, k+ 2n ] with (r a (x ( dk cos x sin x dx = 2r ( cos x sin x dx. c k a n (x = and a n (x = a n (x for x [, 2. Since k+ 2n k 2n { r if x 2n k= [c k, d k, r elsewhere in [,, a n (x ( cos x sin x dx = k+ 2n k 2n a (x ( cos x sin x dx, k+ 2n ]} 2n k=.

12 holds, the conditions in (3 and (4 are satisfied. The r and x remain and since a n (x = r, the curve x n (ϕ = x + (r a n (s ( sin s cos s ds consists of circular arcs all with the same curvature 2r. It remains to show the convergence. Since x n (ϕ and x (ϕ coincide for ϕ = k 2n, we find for ϕ [ k k+ 2n, 2n ], using a n (x, a (x r, that x n (ϕ x (ϕ 4r ds 2r k 2n n, which shows the convergence in C [, 2]. Our third result concerns an approximation by smooth sets of constant width. On page 6 of [8] Chakerian and Groemer write that Tanno in [2] in 976 was the first to prove that arbitrary sets of constant width in 2d can be approximated by sets of constant width with smooth boundaries. Using (6 we can give a direct proof. Corollary 9 Let γ [,. Any set of constant width in 2 dimensions can be approximated in C,γ -sense by sets of constant width with C -boundary. Proof. Suppose that a, r and x as in Recipe 3 describes the boundary of the set of constant width G. Let ζ ε be the usual mollifier: { ( ζ ε (s = cε exp ε 2 s 2 ε for s < ε 2 with ζ (s ds = c. for s ε We replace a by where a ε = ζ ε a 2 sin, ζ ε a sin 2 cos, ζ ε a cos, (34 f, g = f (x g (x dx, r by r ε = max (r, a ε and x by x ε + ( r ε r. The construction in (34 implies that (3 and (4 are satisfied. Since a L (R one finds a L p (, 2 for all p < and ζ ε a a L p (,2 for ε. Hence, also a ε a Lp (,2 for ε. Moreover ζ ε a a and hence with the L -convergende of a ε to a, we also find a ε a for ε. Comparing x (ϕ = x + (r a (s ( sin s cos s ds with the ε-perturbed version, one finds by taking p large enough for p > γ to hold, that by a Sobolev imbedding which completes the proof. x (ϕ x ε (ϕ C,γ [,2] c p,γ x (ϕ x ε (ϕ W,p [,2] c p,γ x x ε + c p,γ a a ε L p (,2 for ε, One year after Tanno s approximation by C functions Weber [2] proved that one may approximate sets of constant width by those that have analytic boundary. Also this result is directly obtained using (6. Corollary Let γ [,. Any set of constant width in 2 dimensions can be approximated in C,γ -sense by sets of constant width with analytic boundary. Proof. If a ε is a smooth function approximating a as in Corollary 9, then, by taking a finite Fourier series a N,ε (x := N k= N 2 e ik, a ε e ikx one finds an analytic approximation of a ε and hence of a. Note that the coefficients for k {, } are zero, so a N,ε satisfies (3 and (4. Moreover, a N,ε a ε can be made arbitrary small by taking N large. We do not get that a N,ε equals a ε = a, but through replacing r by r + a N,ε a ε when r = a, one can still satisfy (5. 2

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