Bárány, Imre

English

Bárány Imre

University of Szeged

Acta Sei. Math., 52 (1988), 93—111 On the minimal ring containing the boundary of a convex body IMRE BÁRÁNY 1. Let .KcR2 be a convex compact set with boundary C. For each point x£K there exist a minimal circular disc B(R(x), x) containing К and a maximal circular disc B(r(x), x) contained in K, where B(r, x) denotes the disc with radius r and cen-ter x. The function R(x)—r(x) attaines its minimal value in a unique point x0£K. This was shown by BONNESEN [1], Bonnesen and FENCHEL [2]. So the circular ring around x0 with radii R(x0) and r(x0), respectively, is the minimal ring containing the boundary С of K. This result was used by Bonnesen and Fenchel [2] to sharpen the isoperimetric inequality in R2. Later I. VINCZE [7] showed that r n min |Д(х): x<jK) f$ R(xo) - 2 and max{/-(x): x£K} W r(xo) and these inequalities are sharp. Answering a question due to I. Vincze we generalize the inequalities (1) and (2) to arbitrary dimension. To do so we need a theorem characterizing the minimal ring in Rd. For d—2 and d= 3 such a theorem was found by Bonnesen [1] and by KRITIKOS [4]. The main tool in the proof of our results is the use of convex analysis (see: Йоффе — Тихомиров [3] and ROCKEFELLAR [5]). 2. Again, let Kc. R"1 be a convex compact set with boundary C. B(r, x) stands for the ball with radius r and center x. For x£K we define R(x) = min {R: B(R, x) ¡5 K), r(x) = max {/•: B(r, x) g A"}. Received January 18, 1985. 94 I. Bárány It is easy to see that the maximum and minimum above exist, so the definition is correct. Moreover, this means that for each x£K there exist points p and q such that p,q£C and [|;c—/>|| = R(;c) and q\\ = r(x). In this case we say that p supports R(x) and q supports r(x). Theorem 1. There exists a point x0(:K in which the function R(x)—r(x) attaines its minimal value. This point x0 is unique. Theset {x€Rd: r ( x 0 ) s | | x — i s called the minimal ring containing C. The characterization theorem for the minimal ring is this: Theorem 2.. The point x0£K is the center of the minimal ring if and only if there are points pi, ...,pk£C supporting R(x0) and qy, ..., qfcC supporting r(xQ) (k, /Si) such that '" t}rwii55r! 1= '1" where conv denotes the convex hull. There is a certain converse to this theorem. We describe it when x 0 =0. Theo rem 3. Letpx, ...,pk, q1} ..., qi be vectors in Rd such that (0 ll/>ill = ... = llAll = Ksr , (ii) M = ... = | |g I | |=»^0, (iii) {pJR: /=1 , ...,k}Dconv {qj/r:j= 1, ...,/} ^0 , (iv) each pt is contained in the halfspaces {x€R": ( q j , q j - x ) ^ 0} ( j = 1, ...,/). In this case there exists a convex compact set Kcz Rd for which R(x)—r(x) attaines its minimal value at x0=0, R(0)=R, r(0)=r and R(0) is supported by plt ...,pk£C and r(0) is supported by q2, ...,qt(iC. Now we give the generalization of the inequalities (1) and (2). Theo rem 4. For d s 3, max r(x)lr(x0) is not bounded from above. On the other hand, for d^3, min R(x)/R(x0) S 1cos2 a0 + cos <x0 - 1 + — - — ) » 0.8054, jL \ COS OCq / where a0£(0, n/2) is the root of the equation sin2 a—2 cos3 a=0 . This inequality is sharp. 3. This section contains the proofs. We start with some simple facts and obser-vations. 95 I. Barany: On the minimal ring containing the boundary of a convex body Cla im 1. R(x) = max.\\x-p\\ = max\\x-p\\, Pfcii pet, r(x) = inf II*-/>11 = min ||x-/>||, p(K pZC and the points in which the maximum (minimum) is attained support R(x) (r(x), re-spectivelyJ. Cla im 2. (a) " P ^ ^ j - ^ C x O + K f e ) ) and if equality holds here, then there is a unique p£C supporting iî((*1+*2)/2) and this point lies on the straight line through xx and x2, and p supports R(xJ and R(X2) as well. (b) r ^ j ë y N + ' W ) P roo f , (a) Let p£C be a point of support for R((xx+x2)/2). Then p£B(R(x1), x1)r\B(R(x2), x2) and the triangle-inequality proves the claim. (b) Obviously conv (B(r(xx), Xx)UB(r(x2), and an easy calculation shows that E conv(B(r(Xl), Xl)UB(r(x2), x2)). P r o o f of T h e o r e m 1. By Claim 2, R(x) is a convex, r(x) is a concave func-tion. So R(x)—r(x) is convex and attaines its infimum. What we have to show is the uniqueness of the minimum. This will be done by showing that x1, x2dK, x19£x2 and R(x1)—r(x1)=R(x^)—r(x2)=h implies that R((xx+x2)/2)— r((xx+x2)j2)<h. Convexity implies that R((x1+x2)/2)—r((x1+x2)/2)^h, so assume, by way of contradiction, that R((x1+x2)l2)—r((x1+x2)l2)=h. Then by Claim 2, we have R((x1+x2)/2)=ll2(R(x1)+R(x2)) and a unique point p£C supporting ^ ( x j , R(xs) and R((x1+x2)/2) a n d p lies on the straight line through xt and x2. Without loss of generality we suppose that x2 lies between Xx and p on this line. By our assumption R(xx)~f(xj)=R(X2)—r(x2), so B(r(x2), x2)QB(r(xi), XX), and then there is a unique point qÇC supporting r(x2) and this point lies on the line segment joining x2 and p. But K contains the set conv (B(r(x{), X])U {/>}) and this set contains q in its interior. This contradicts the assumption 96 I. Bárány For fixed p£C define Z(p) as the set of unit outer normals to K at p, i.e., Z(p) = {zdRd: ||z|| = 1 ,{z,p) = max(z, x)}. Define now r = {(p,z)£RdxRd: z£Z(p)}. It is clear that T is compact. Claim 3. (a) R(x) — max {(z,p—x): {p,z)^T}, (b) r(x)=min {(z,p-x): (p,z)£T}. Proof , (a) Clearly for each (p,z)£T (z,p-x) =S | |z|H|/?-x| | = ||/>-x|| == R(x). If p0 supports R(x), then (p0, ((p0-x)l\\pa-x\\)£r and ( l ^ r l I T , P o - x ) = R(x). (b) Trivially (z, p—x)sr(x) for each (p, z)£T. On the other hand it is easy to check that if p0 supports r(x), then Z(p0)={p0—xl\\p0—x\\} and ( l = D Using Claim 3 the function r: K^-R1 can be extended over the whole space Rd. It is again easy to see that the extended r(x) is concave, and so the function R(x)—r(x) (x£Rd) attaines its minimal value at x0£K only. To prove Theorem 2 we need some definitions and theorem from convex analysis. Def in i t ion . Let f: Rd—R be a convex function. The set df(x) = {x*£Rd: <x*,z-x> s / ( z ) - / ( x ) (for every z£Rd)} is the subgradient of f at x. It is well-known that the subgradient of a finite convex function is nonempty, convex and compact. Theorem A (Fenchel, Rockafellar—Moreau, see [5]). Let f : Rd—R be con-vex, g: Rd—R concave functions, finite over the whole space. Then f(x)—g(x) attains its minimum at x0 if and only if 0 idf{x0) + d(-g)(x0). Here the last addition is meant in the Minkowski sense; (—g) is a convex func-tion so d(—g)(x0) is its subgradient at x0. 97 I. Barany: On the minimal ring containing the boundary of a convex body Theorem В (Йоффе — Тихомиров [3]). Assume Г is compact and the map y~-{x*, fly)6R'lXR is continuous. Let f(x)=sup {(x*, x)+ar: y€f}. Then f : Rd—R is a finite convex function and r)/(x0)=conv {x*: у£Г and (x*, x0)+ay=f(x0)}. Now we are ready to prove Theorem 2. P roo f of T h e o r e m 2. First by Theorem В dR(x0) = conv { - z : (p, z)£T, <z,p-x0> = Л(х0)}, d(-r)(x0) = conv {z: (p, z)€T, (z,p-x„> = r(x0)j. By Theorem A, R(x)—r(x) is minimal at x„ if a n d only if for some x*€ Rd, x*£dR(x0) к and —x*€#(—r)(x0). But x*£dR(x0) is the same as x*= — ^aizi for some i=l к a^O, 2, a ; = 1 and z; with (pt, zt)£T, (zi,pi-x0)=R(x0). i This is true if and only if z—/»;—x0/||p,— x0||, i.e., if pt supports R(x0). Simi-i i larly — x*$D(— r)(x0) is equivalent to — x*= 2 PJWJ f ° r some fij^Q, 2 Pj— 1 and j=i i Wj v/itb(qj,Wj)£r, (Wj,qj-x0)=r(x0). In this case, again Wj=(qj-x0)l\\qJ—x0\\ and qs supports r(jc0). These conditions imply that R(x)—r(x) is minimal at x0 if and only if there exist points p1,...,pk£C supporting R(x0) and qx, ..., qt£C supporting r(x0) such that 'И-So we are finished with the proof. We mention that к— 1 (or 1= 1) implies that К is a ball. Further, it can be shown that if conv Pflconv Q ^ 0 for some P,Q£ Rd, then there are subsets P'QP and Q'QQ such that conv P'f lconv Q V 0 and \P'\ + \Q\'sd+2. This means that we can suppose k+l^d+2 in Theorem 2. I mention here that the "only if" part of Theorem 2 can be proved in a simpler way: Set P={(pi-x0)/R(x0): i= 1, ...,*} and Q = {(qs-x0)/r(x0): y'= 1,.. . , /}. If conv Pflconv Q—0, then there is a hyperplane separating P and Q with normal fl€Rd, say. One can easily see that R(x0)>R(x0+a) and r(x0)<r(x0+a) which shows that R(x)—r(x) cannot attain its minimal value at x0. P roo f of T h e o r e m 3. Set *min = СОПУ (В(Г, 0) U {P1, ..., Pk}). K^ = B(R ,0 )n П {*•• <qj, gj-x) s o}. J=1 98 I. Bárány It is easy to see that both Kmin and Kmax satisfy the conditions of Theorem 2 with x 0 = 0 and Pi, ...,pk, <7i, ..., q,. Moreover, any convex compact set K with KmiaQ QKQ Kmux will do the same. P roo f of Theorem 4. First part. We construct a convex compact set K<z Rd for each d^ 3 such that max r(x)[r(xQ) is "large". Let Pi, p2,q!, q2 be the vertices of a square such that \\pL\\ = || j?2|| = ll?ill = ll^ll = 1 and the length of the diagonals pxp2 and qxq2 is 2—e (where e > 0 is small). The hy-perplanes (q!,qi—x)=0 and (q2, q2—x)=0 meet in an affine flat A. The halflines starting from the origin in directions py and p2 meet A in the points p^Rpi and p2—Rp2• Consider the set ATmM from Theorem 3 with Pi, p2 and qx,q2. A simple calculation shows that (2 / 2 \ — e — , r(0) = 1, and maxr(x) = So we have maxr(x) _ ( eM - 1 '2 r(xo) " r ~ T j which indeed tends to infinity as e—0. Second part. Let J£czRd (d^ 3) be convex compact body and suppose that R(x)—r(x) attaines its minimal value at 0 and r(x0)= 1, R(x0)—R. By Theorem 2 there exist points px, ...,Pi supporting R(x0) and q±, ...,qt supporting r(x0) with com{pi/R: i = 1, ..., fc}Plconv {q}\ j = 1, ..., /} ^ 0, and we may assume k, / s 2 , k+l^d+2. Then com {px, ...,pk} is a simplex whose nearest point to the origin is p0 say. Clearly ¡Pi—Poll — •••=11 Pk—Po\\ and the angle between the vectors pt and p0 is the same for each z. Denote this angle by a. Now the halfspaces {q3,q}—x)^0 (j=\, ...,l) have to contain the simplex conv ...,pk} and so the point p0 as well. On the other hand, for some j— 1, ..., I the angle between the vectors qs and p0 is not larger than a for otherwise conv{pi/R: i = 1, ..., k}Dconv {qj: j = 1, ...,/} = 0. This implies that 0 S (qs, qj -p0) - 1 - (qj, p0) = = 1 - M • llA.ll cos (<.qjOp0) 1 - R cos2 a. Consider now min R(x)= Q and set R(x)— g, x£K. Then B(g, x) contains the points pi, ...,pk and the ball B(l, 0), so it contains the point p0— —Po/llPoll a s well. We are going to give an estimation from below for the radius of the smallest ball containing the points p 0 , p t , . . . ,pk . It is clear that the smallest ball containing 99 I. Barany: On the minimal ring containing the boundary of a convex body Pi, ...,pk is B(Rsin a,p0) and so R sin k^q. However if ||p0—/»0[| = 2? cos a + l > >/? sin a, then B(R sin a,p0) does not contain p0. In this case, using some elemen-tary geometry, we get the estimation Define now 1 + 2.R cos g + R2 ' - 2(1 + /?cos a) ' sin a if Ps ina s -Rcosa + l, l+2 i?cosa + R2 otherwise 2/?(l + cos a) where K s l , 0Sa^7t /2 and i?cos2a=Sl. What we have to do is to find the minimum o f / in the domain determined by these inequalities. This is a routine calculation. The main steps are: 1) for R fixed f(R, a) is monotone non-decreasing, so the minimum is attained on the curve R cos2 a = 1; 2) on this curve the minimum of / is equal to j (cos2 oc0 + cos Oo — 1 + cos a0) where a0 is the solution of the equation sin2 a—2 cos3 a = 0 with 0^a0^7r/2. This proves that t(*) 1 ( 2 , 1 1 S cos2a0 + c o s a 0 - l + . o) 2 \ cos a0 ) (A\ min R(x) Finally we give an example showing that equality can occur here for d= 3, 4, . . . . Again, let pi, p2, qx, q2 be the vertices of a square such that the diagonals plt p2 and qu q2 meet in a point q and the angle between q and p1,p2,q1, q2 equals a0 . Now set p= cos - 2 a0 px and p2= cos - 2 a0 p2 and apply Theorem 3 with the vectors Pi,Pz, q2 to get the convex compact set Kmtn. An easy calculation shows that for A"mln (4) holds with equality. Acknowledgement. The author is indebted to professor I . VINCZE for raising the problem and for fruitful discussions. • T 100 I. Barany: On the minimal ring containing the boundary of a convex body References [1] Т. BONNESEN, Über das isoperimetrische Defizit ebener Figuren, Math. Ann., 91 (1924), 252—268. [2] T. BONNESEN and W. FENCHEL, Theorie der konvexen Körper, de Gruyter (Berlin, 1956). [3] А. Д. Йоффе, В. M. Т и х о м и р о в , Теория экстремальных задач, Наука (Москва, 1974). [4] N. KRITIKOS, Über konvexe Flachen und einschließende Kugeln, Math. Ann. 96 (1927), 583— 586. [5] T. R. ROCKAFELLAR, Convex Analysis, Princeton (1970). [6] J. SZŐKEFALVI-NAGY, Konvexe Kurven und einschließende Kreisringe, Acta. Sei. Math. (Szeged), 10 (1941—1943), 174—184. [7] ST. VINCZE, Über den Minimalkreisring einer Eilinie, Acta Sei. Math. (Szeged), 11 (1947), 133— 138. [8] I. VINCZE, Über Kreisringe, die eine eilinien einschließen, Studia Sei. Math. Hungar, 9 (1974), 155—159. INSTITUTE OF MATHEMATICS OF THE HUNGÁRIÁN ACADEMY OF SCIENCES REÁLTANODA U. 13—15 1053 BUDAPEST, HUNGARY 

On the minimal ring containing the boundary of a convex body

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On the minimal ring containing the boundary of a convex body

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