On the Circle Covering Theorem by A.W. Goodman and R.E. Goodman

Discrete & Computational Geometry, Mar 2017

In 1945, A.W. Goodman and R.E. Goodman proved the following conjecture by P. Erdős: Given a family of (round) disks of radii \(r_1\), \(\ldots \), \(r_n\) in the plane, it is always possible to cover them by a disk of radius \(R = \sum r_i\), provided they cannot be separated into two subfamilies by a straight line disjoint from the disks. In this note we show that essentially the same idea may work for different analogues and generalizations of their result. In particular, we prove the following: Given a family of positive homothetic copies of a fixed convex body \(K \subset {\mathbb {R}}^d\) with homothety coefficients \(\tau _1, \ldots , \tau _n > 0\), it is always possible to cover them by a translate of \(\frac{d+1}{2}\big (\sum \tau _i\big )K\), provided they cannot be separated into two subfamilies by a hyperplane disjoint from the homothets.

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On the Circle Covering Theorem by A.W. Goodman and R.E. Goodman

Arseniy Akopyan 0 1 2 3 Alexey Balitskiy 0 1 2 3 Mikhail Grigorev 0 1 2 3 Editor in Charge: János Pach 0 Moscow Institute of Physics and Technology , Institutskiy per. 9, 141700 Dolgoprudny , Russia 1 Institute of Science and Technology Austria (IST Austria) , Am Campus 1, 3400 Klosterneuburg , Austria 2 Department of Mathematics, Massachusetts Institute of Technology , 182 Memorial Dr., Cambridge, MA 02142 , USA 3 Institute for Information Transmission Problems RAS , Bolshoy Karetny per. 19, 127994 Moscow , Russia In 1945, A.W. Goodman and R.E. Goodman proved the following conjecture by P. Erd o˝s: Given a family of (round) disks of radii r1, . . ., rn in the plane, it is always possible to cover them by a disk of radius R = ri , provided they cannot be separated into two subfamilies by a straight line disjoint from the disks. In this note we show that essentially the same idea may work for different analogues and generalizations of their result. In particular, we prove the following: Given a family of positive homothetic copies of a fixed convex body K ⊂ Rd with homothety coefficients τ1, . . . , τn > 0, it is always possible to cover them by a translate of d +21 τi K , provided they cannot be separated into two subfamilies by a hyperplane disjoint from the homothets. Goodman-Goodman theorem; Non-separable family; Positive homothets - Mathematics Subject Classification 52C10 · 52C17 1 Introduction Consider a family K of positive homothetic copies of a fixed convex body K ⊂ Rd with homothety coefficients τ1, . . . , τn > 0. Following Hadwiger [6], we call K non-separable if any hyperplane H intersecting conv K intersects a member of K. Answering a question by Erdo˝s, A.W. Goodman and R.E. Goodman [4] proved the following assertion: Theorem 1.1 (A.W. Goodman, R.E. Goodman, 1945) Given a non-separable family K of Euclidean balls of radii r1, . . . , rn in Rd , it is always possible to cover them by a ball of radius R = ri . Let us outline here the idea of their proof since we are going to reuse it in different settings. First, A.W. Goodman and R.E. Goodman prove the following lemma, resembling the 1-dimensional case of the general theorem: Lemma 1.2 Let I1, . . . , In ⊂ R be segments of lengths 1, . . . , n with midpoints c1, . . . , cn. Assume the union Ii is a segment (i.e. the family of segments is nonseparable). Then the segment I of length i with midpoint at the center of mass c = i cii covers Ii . Next, for a family K = {oi + ri B} (B denotes the unit ball centered at the origin of Rd ), A.W. Goodman and R.E. Goodman consider the point o = riroii (i.e., the center of mass of K if the weights of the balls are chosen to be proportional to the radii). They project the whole family onto d orthogonal directions (chosen arbitrarily) and apply Lemma 1.2 to show that the ball of radius R = ri centered at o indeed covers K. In [2], K. Bezdek and Z. Lángi show that Theorem 1.1 actually holds not only for balls but also for any centrally-symmetric bodies: Theorem 1.3 (K. Bezdek and Z. Lángi, 2016) Given a non-separable family of homothets of centrally-symmetric convex body K ⊂ Rd with homothety coefficients τ1, . . . , τn > 0, it is always possible to cover them by a translate of τi K . The idea of their proof is to use Lemma 1.2 to deduce the statement for the case when K is a hypercube, and then deduce the result for sections of the hypercube (which can approximate arbitrary centrally-symmetric bodies). It is worth noticing that Theorem 1.3 follows from Lemma 1.2 by a more direct argument (however, missed by A.W. Goodman and R.E. Goodman). In 2001, F. Petrov proposed a particular case of the problem (when K is a Euclidean ball) to Open Mathematical Contest of Saint Petersburg Lyceum N−o 239 [1]. He assumed the following solution (working for any symmetric K as well): For a family K = {oi +τi K }, consider a homothet τi K +o with center o = τiτoii . If τi K +o does not cover K, then there exists a hyperplane H separating a point p ∈ conv K \ τi K + o from τi K + o . Projection onto the direction orthogonal to H reveals a contradiction with Lemma 1.2. Another interesting approach to Goodmans’ theorem was introduced by K. Bezdek and A. Litvak [3]. They put the problem in the context of studying the packing analogue of Bang’s problem through the LP-duality, which gives yet another proof of Goodmans’ theorem for the case when K is a Euclidean disk in the plane. One can adapt their argument for the original Bang’s problem to get a “dual” counterpart of Goodmans’ theorem. We discuss this counterpart and give our proof of a slightly more general statement in Sect. 4. The paper is organized as follows. In Sect. 2 we prove a strengthening (with factor d +21 instead of d) of the following result of K. Bezdek and Z. Lángi: Theorem 1.4 (K. Bezdek and Z. Lángi, 2016) Given a non-separable family of positive homothetic copies of a (not necessarily centrally-symmetric) convex body K ⊂ Rd with homothety coefficients τ1, . . . , τn > 0, it is always possible to cover them by a translate of d τi K . 2 A Goodmans-Type Result for Non-symmetric Bodies Let K ⊂ Rd be a (not necessarily centrally-symmetric) convex body containing the origin and let K ◦ = { p : p, q ≤ 1 for all q ∈ K } (where ·, · stands for the standard inner product) be its polar body. We define the following parameter of asymmetry: min min {μ > 0 : (K − q) ⊂ −μ(K − q)}. σ = q∈int K It is an easy exercise in convexity to establish that min{μ > 0 : (K − q) ⊂ −μ (K − q)} = min {μ > 0 : (K − q)◦ ⊂ −μ(K − q)◦}. So an equivalent definition (which is more convenient for our purposes) is min min {μ > 0 : (K − q)◦ ⊂ −μ(K − q)◦}. σ = q∈int K The value σ1 is often referred to as Minkowski’s measure of symmetry of body K (see, e.g., [5]). Theorem 2.1 Given a non-separable family of positive homothetic copies of (not necessarily centrally-symmetric) convex body K ⊂ Rd with homothety coefficients Fig. 1 Illustration of the proof of Theorem 2.1 Proof We start by shifting the origin so that K ◦ ⊂ −σ K ◦. For a family K = {oi + τi K }, consider the homothet σ +21 τi K + o with center o = τiτoii . Assume that σ +21 τi K + o does not cover K, hence there exists a hyperplane H (strictly) separating a point p ∈ conv K \ σ +21 τi K + o from σ +21 τi K + o . Consider the orthogonal projection π along H onto the direction orthogonal to H . Suppose the segment π(K ) is divided by the projection of the origin in the ratio 1 : s. Since K ◦ ⊂ −σ K ◦, we may assume that s ∈ [1, σ ]. Identify the image of π with the coordinate line R and denote Ii = [ai , bi ] = π oi + τi K , ci = π(oi ), i = bi − ai , L = i (see Fig. 1). Note that the i are proportional to the τi , and that s(ci −ai ) = bi −ci . Denote c = π(o) = Li ci and I = [a, b] = π σ +21 τi K +o the segment of length σ +21 L divided by c in the ratio 1 : s. Also consider the midpoints ci = ai +2bi . By Lemma 1.2, the segment I = [a , b ] of length L with midpoint at c = Li ci covers the union Ii = π(K). Let us check that I ⊂ I , which would be a contradiction, since π( p) ∈ I , π( p) ∈/ I . First, notice that ci = = ci , 1 1 1 a = c − 2 L ≥ c − 2 L ≥ c − 1 + s L = a. Fig. 2 Illustration of the proof of Lemma 2.2 ci − ci = 1 1 s − 1 b = c + 2 L = c + (c − c) + 2 L = c + 2(s + 1) s − 1 1 s σ + 1 ≤ c + 2(s + 1) L + 2 L ≤ c + 1 + s 2 L = b. Lemma 2.2 (H. Minkowski and J. Radon) Let K be a convex body in Rd . Then σ ≤ d, where σ denotes the parameter of asymmetry of K , defined above. For the sake of completeness we provide a proof here. Proof Suppose the origin coincides with the center of mass g = K x d x / K d x . We show that K ◦ ⊂ −d K ◦. Consider two parallel support hyperplanes orthogonal to one of the coordinate axes O x1. We use the notation Ht = {x = (x1, . . . , xd ) : x1 = t } for hypeplanes orthogonal to this axis. Without loss of generality, these support hyperplanes are H−1 and Hs for some s ≥ 1. We need to prove s ≤ d. Assume that s > d. Consider a cone C defined as follows: its vertex is chosen arbitrarily from K ∩ Hs ; its section C ∩ H0 = K ∩ H0; the cone is truncated by H−1. Since C is a d-dimensional cone, the x1-coordinate of its center of mass divides the segment [−1, s] in ratio 1 : d. Therefore, the center of mass has positive x1-coordinate. It follows from convexity of K that C \ K lies (non-strictly) between H−1 and H0, hence the center of mass of C \ K has non-positive x1-coordinate. Similarly, K \ C lies (non-strictly) between H0 and Hs , hence its center of mass has non-negative x1coordinate. Thus, the center of mass of K = (C \ (C \ K )) ∪ (K \ C ) (see Fig. 2) must have positive x1-coordinate, which is a contradiction. Corollary 2.3 The factor d in Theorem 1.4 can be improved to d +21 . Proof The result follows from Theorem 2.1 and Lemma 2.2. An alternative proof of this corollary that avoids Lemma 2.2 is as follows. We use the notation of Theorem 1.4. Consider the smallest homothet τ K , τ > 0, that can cover K (after a translation to τ K + t , t ∈ Rd ). Since it is the smallest, its boundary touches ∂ conv K at some points q0, . . ., qm (m ≤ d) such that the corresponding support hyperplanes H0, . . ., Hm bound a nearly bounded set S, i.e., a set that can be placed between two parallel hyperplanes. Circumscribe all the bodies from the family K by the smallest homothets of S and apply Theorem 2.1 for them (note that if m < d then S is unbounded, but that does not ruin our argument). Since S is a cylinder based on an m-dimensional simplex, its parameter of asymmetry equals m ≤ d, and we are done. Remark 2.4 Up to this moment the best possible factor for non-symmetric case is unknown. Bezdek and Lángi [2] give a sequence of examples in Rd showing that it is impossible to obtain a factor less than 23 + 3 √23 (> 1) for any d ≥ 2. 3 A Sharp Goodmans-Type Result for Simplices Consider the case when K ⊂ Rd is a simplex. In this section we are only interested in separating hyperplanes parallel to a facet of K . Theorem 3.1 Let K be a family of positive homothetic copies of a simplex K ⊂ Rd with homothety coefficients τ1, . . . , τn > 0. Suppose any hyperplane H (parallel to taofcaocveetrof KK) ibnytearstercatninsglatceonovf d +2K1 inteτriseKct.s Ma omreeomvbeer,r tohfe Kfa.cTtohrend +2it1 iscapnonsostibblee improved. Proof A proof of possibility to cover follows the same lines as (and is even simpler than) the proof of Theorem 2.1. Let K have its center of mass at the origin. For a family K = {oi + τi K }, consider a homothet d +21 Assuming d +21 τi K + o does not cover K, we find a hyperplane H (strictly) separating a point p ∈ conv K \ d +21 τi K + o from d +21 τi K + o . Note that H can be chosen among the hyperplanes spanned by the facets of d +21 τi K + o , so H is parallel to one of them. After projecting everything along H onto the direction orthogonal to H , we repeat the same argument as before and show that (in the notation from Theorem 2.1) which contradicts our assumption. Next, we construct an example showing that the factor d +21 cannot be improved. Consider a simplex K = x = (x1, . . . , xd ) ∈ Rd : xi ≥ 0, i=1 xi ≤ where N is an arbitrary large integer. Section it with all hyperplanes of the form {xi = t } or of the form id=1 xi = t (for t ∈ Z). Consider all the smallest simplices generated by these cuts and positively homothetic to K . We use coordinates Fig. 3 Example for d = 2 and N = 5 i=1 bi ≤ d(d + 1) N , 2 to denote the simplex lying in the hypercube {bi ≤ xi ≤ bi + 1, i = 1, . . . , d}. For d = 2 (see Fig. 3) we compose K of the simplices with the following coordinates: For d = 3: It is rather straightforward to check that each bi ranges over the set {0, 1, . . . , d N }, and their sum is not greater than d(d2+1) N . Therefore, the chosen family K is indeed non-separable by hyperplanes parallel to the facets of K . Moreover, the chosen simplices touch all the facets of K , so K is the smallest simplex covering K. Finally, we note that any one-dimensional parameter of K (say, its diameter) is d2((dd+N1+)1N) times greater than the sum of the corresponding parameters of the elements of K, and this ratio tends to d+1 as N → ∞. 2 4 A “Dual” Version of Goodmans’ Theorem Lemma 4.1 Let I1, . . . , In ⊂ R be segments of lengths 1, . . . , n with midpoints c1, . . . , cn. Assume every point on the line belongs to at most k of the interiors of the Ii . Then the segment I of length k1 i with midpoint at the center of mass c = i cii Proof Mark all the segment endpoints and subdivide all the segments by the marked points. Next, put the origin at the leftmost marked point and numerate the segments between the marked points from left to right. We say that the i -th segment is of multiplicity 0 ≤ ki ≤ k if it is covered ki times. We keep the notation Ii for the new segments with multiplicities, ci for their midpoints, and i for their lengths. Note that the value i ci is preserved after this change of notation: it is the coordinate of the i center of mass of the segments regarded as solid one-dimensional bodies of uniform density. Note that ci = 1 + · · · + i−1 + 21 i . We prove that c = (this would mean that the left endpoint of I is contained in conv endpoint everything is similar). The inequality in question Ii ; for the right 2 , which is true, since k ≥ ki . Theorem 4.2 Let k be a positive integer, and K be a family of positive homothetic copies (with homothety coefficients τ1, . . . , τn > 0) of a centrally-symmetric convex body K ⊂ Rd . Suppose any hyperplane intersects at most k interiors of the homothets. Then it is possible to put a translate of k1 τi K into their convex hull. Proof As usual, for a family K = {oi + τi K }, consider a homothet k1 with center o = τiτoii . Assume k1 τi K + o does not fit into conv K, then there exists a hyperplane H separating a point p ∈ k1 τi K + o from conv K. After projecting onto the direction orthogonal to H , we use Lemma 4.1 to obtain a contradiction. Remark 4.3 The estimate in Theorem 4.2 is sharp for any k, as can be seen from the example of k translates of K lying along the line so that consecutive translates touch. Acknowledgements Open access funding provided by Institute of Science and Technology (IST Austria). The authors are grateful to Rom Pinchasi and Alexandr Polyanskii for fruitful discussions. Also the authors thank Roman Karasev, Kevin Kaczorowski, and the anonymous referees for careful reading and suggested revisions. The research of the first author is supported by People Programme (Marie Curie Actions) of the European Union’s Seventh Framework Programme (FP7/2007-2013) under REA grant agreement n◦[291734]. The research of the second author is supported by the Russian Foundation for Basic Research Grant 15-01-99563 A and Grant 15-31-20403 (mol_a_ved). Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. 1. Berlov , S. , Ivanov , S. , Karpov , D. , Kokhas' , K. , Petrov , F. , Khrabrov , A. : Problems from St . Petersburg School Olympiad on Mathematics 2000 - 2002 . Nevskiy Dialekt, St. Petersburg ( 2006 ) 2. Bezdek , K. , Lángi , Z. : On non-separable families of positive homothetic convex bodies . Discrete Comput. Geom . 56 ( 3 ), 802 - 813 ( 2016 ). doi:10.1007/s00454- 016 - 9815 -1 3. Bezdek , K. , Litvak , A.E. : Packing convex bodies by cylinders . Discrete Comput. Geom . 55 ( 3 ), 725 - 738 ( 2016 ). doi:10.1007/s00454- 016 - 9760 -z 4. 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On the Circle Covering Theorem by A.W. Goodman and R.E. Goodman, Discrete & Computational Geometry, 2017, DOI: 10.1007/s00454-017-9883-x