NOTE. On the Maximum Number of Touching Pairs in a Finite Packing of Translates of a Convex Body

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1 Journal of Combinatorial Theory, Series A 98, (00) doi: /jcta , available online at on NOTE On the Maximum Number of Touching Pairs in a Finite Packing of Translates of a Convex Body Károly Bezdek Department of Geometry, Eötvös University, Pázmány Péter sétány 1/c, H-1117 Budapest, Hungary kbezdek@ludens.elte.hu, bezdek@cs.elte.hu Communicated by Victor Klee Received January 15, 001 A Minkowski space M d =(R d, ) is just R d with distances measured using a norm. A norm is completely determined by its unit ball {x R d x [ 1} which is a centrally symmetric convex body of the d-dimensional Euclidean space E d. In this note we give upper bounds for the maximum number of times the minimum distance can occur among n points in M d,d\ 3. In fact, we deal with a somewhat more general problem namely, we give upper bounds for the maximum number of touching pairs in a packing of n translates of a given convex body in E d,d\ Elsevier Science (USA) /0 $ Elsevier Science (USA) All rights reserved. 1. INTRODUCTION Let K be a convex body, i.e., a compact convex set with non-empty interior in the d-dimensional Euclidean space E d. Harborth [10] proved the remarkable result that the maximum number of touching pairs in a packing of n congruent circles in E (i.e. in a family of n non-overlapping congruent circles in E ) is precisely N3n `1n 3M. (See [11] for further discussions on this problem). Recently the same but much harder question has been answered for packings of congruent circles in the hyperbolic plane by Bowen and Radin in the elegant papers [5, 6]. Moreover, with the help of the properly modified method of [10] Brass [7] could prove that the maximum number of touching pairs in a packing of n translates of a convex body K in E is N3n `1n 3M, ifk is not a parallelogram, and N4n `8n 1M, ifk is a parallelogram. Agarwal and Pach [1, p. 11] raise the higher dimensional analogue question calling the attention to the 19

2 NOTE 193 fact that besides the following trivial upper bound nothing seems to be known about this problem: Let K be a convex body in E d,d\ 3 and let H(K) be the Hadwiger number of K which is defined as the largest number of non-overlapping translates of K that can touch K. Then the maximum number of touching pairs in a packing of n translates of the convex body K in E d (i.e. in a family of n non-overlapping translates of the convex body K in E d ) is at most (H(K)/) n. In this note we improve this estimate as follows. Let d(k) be the density of a densest packing of translates of the convex body K in E d,d\ 3. Moreover, let h(k) be the one-sided Hadwiger number of K which has been introduced in [3] as the maximum number of non-overlapping translates of K that can touch K such that their convex hull does not contain K in its interior. Then let Iq(K)=(Svol d 1 (bdk)) d / (Vol d (K)) d 1, which is very often called the isoperimetric quotient of the convex body K in E d, where Svol d 1 (bdk) denotes the (d 1)-dimensional surface volume of the boundary bdk of K and Vol d (K) denotes the d-dimensional volume of K. Moreover, let B denote the closed d-dimensional ball of radius 1 centered at the origin o in E d. Finally, if X, Y are subsets of E d and l is a real number, then X+Y={x+y x X, y Y}, X Y={x y x X, y Y} and lx={lx x X}. Now, our result can be phrased as follows. Theorem. Let K be a convex body in E d,d\ 3 and let K o = 1 (K+ ( K)) be the normalized difference body of K, which is centrally symmetric about the origin o of E d. Then the maximum number of touching pairs in a packing of n>1translates of the convex body K in E d,d\ 3 is at most H(K o ) n 1 d (d 1)/d (d(k o 1 )) (H(K o ) h(k o ) 1). Iq(B) 1/d n (d 1)/d Iq(K o ) We remark that due to the Minkowski difference body method H(K)= H(K o ) and h(k)=h(k o ) hold for any convex body K in E d (see [1], [13] and also the next section of this paper for a description of this method). Moreover, it is known ([9]) that H(K) [ 3 d 1 for any convex body K of E d and it has been recently proved in [3] that h(k) [ 3 d 1 1 holds for any convex body K of E d. (The inequality 0 [ H(K) h(k) 1 is rather trivial, where K denotes an arbitrary convex body of E d.) A Minkowski space M d =(R d, ) is just R d with distances measured using a norm. The Theorem has the following corollary.

3 194 NOTE Corollary 1. The maximum number of times the minimum distance can occur among n>1points in M d,d\ 3 is at most 3 d 1 d+1 n(d 1)/d, where w d =p d/ /C( d +1) is the volume of a d-dimensional ball of radius 1 in E d. Moreover, if M d l. denotes R d with the maximum norm, then the maximum number of times the minimum distance can occur among n>1 points in M d l.,d\ 3 is at most 3 d 1 d+1 n(d 1)/d d+1 (3 d 1 1). Finally, we call the reader s attention to Corollary of Section that suggests to compute d(k o ) via the densities of finite packings of translates of the o-symmetric convex body K o in special non-convex containers of E d. Let. PROOF OF THE THEOREM {c 1 +K, c +K,..., c n +K} be an arbitrary packing of n>1 translates of the convex body K in E d,d\ 3 with Then {c 1, c,...,c n }=C n. (c 1 +K) (c +K) (c n +K)={c 1, c,...,c n }+K=C n +K. We begin with the so-called difference body method of Minkowski [1, 13]: If t 1 +K and t +K are two translates of the convex body K, then they are non-overlapping (resp., touching) if and only if t 1 +K o and t +K o are non-overlapping (resp., touching), where K o = 1 (K+( K)). From this fact it follows that {c 1 +K o, c +K o,...,c n +K o } is a packing of n translates of the convex body K o moreover, the number of touching pairs in the packing {c 1 +K o, c +K o,...,c n +K o } is equal to the number of touching pairs in the packing {c 1 +K, c +K,..., c n +K}. Thus, it is sufficient to give an upper bound for the number of touching pairs in the packing {c 1 +K o, c +K o,...,c n +K o }.

4 For our proof of the Theorem we need the following Lemma, which is a stronger version of Theorem 3.1 in [4] and implies Corollary below. The following proof of the Lemma is a somewhat corrected and modified version of the proof of Theorem 3.1 in [4]. Lemma. NOTE 195 Vol d (C n +K o ) Vol d (C n +K o ) [ d(k o). Proof. that (1) Assume that the claim is not true. Then there is an E >0 such Vol d (C n +K o )=n Vol d(k o ) E. d(k o ) Let L be a packing lattice of C n +K o such that C n +K o is contained in the foundamental parallelotope P of L. For each l >0 let Q l denote the d-dimensional cube of edge length l centered at the origin o of E d having edges parallel to the corresponding coordinate axes of E d. Obviously, there is a constant m >0depending on P only such that for each l >0there is a subset L l L with Q l L l +P and L l +P Q l+m. Moreover, let P m (K o ) denote the family of all possible packings of m>1 translates of the o-symmetric convex body K o in E d. The definition of d(k o ) (see [13]) implies that for each l >0 there exists a packing in the family P m(l) (K o ) with centers at the points of C m(l) such that and lim l Q. C m(l) +K o Q l m(l) Vol d (K o ) =d(k Vol d (Q l ) o ). As lim l Q. (Vol d (Q l+m )/Vol d (Q l ))=1 there exist t >0 and a packing in the family P m(t) (K o ) with centers at the points of C m(t) and with C m(t) + K o Q t such that () Vol d (P) d(k o ) < m(t) Vol d(k o ) Vol d (P)+E Vol d (Q t+m ) and < n Vol d(k o ) card(l t ). Vol d (Q t+m ) n Vol d (K o ) Vol d (P)+E

5 196 NOTE Now, for each x P we define a packing of n(x) translates of the o-symmetric convex body K o in E d with centers at the points of C n(x) ={x+l t +C n } {y C m(t) y x+l t +C n +int(k o )}. Obviously, C n(x) +K o Q t+m. Now, in order to compute > x P n(x)dx, for each point y C m(t) we introduce the function q y defined as follows: q y (x)=1 if y x+l t +C n +int(k o ) and q y (x)=0 for any other x P. Then it is easy to see that F n(x)dx=f x P x P 1 n card(lt )+ C q y y C m(t) (x) dx =n Vol d (P) card(l t )+m(t)(vol d (P) Vol d (C n +K o )). Hence, there is a point p P with n(p) \ m(t)1 1 Vol d (C n +K o ) Vol d (P) +n card(lt ) and so (3) n(p) Vol d (K o ) Vol d (Q t+m ) \ m(t) Vol d(k o ) Vol d (Q t+m ) 1 1 Vol d (C n +K o ) Vol d (P) + n Vol d(k o ) card(l t ). Vol d (Q t+m ) Now, (1) implies in a straightforward way that (4) Vol d (P) d(k o ) Vol d (P)+E 1 1 Vol d (C n +K o ) Vol d (P) + n Vol d (K o ) Vol d (P)+E =d(k o). Thus, (), (3), and (4) yield that n(p) Vol d (K o ) Vol d (Q t+m ) > d(k o). As C n(p) +K o Q t+m this contradicts the definition of d(k o ). This completes the proof of the Lemma. L We mention the following immediate corollary of the Lemma which is a stronger analogue of Corollary 3.1 of [4] and connects finite packings of translates of the o-symmetric convex body K o in special non-convex containers of E d to infinite packings of translates of K o in E d via their densities.

6 Corollary. Let P n (K o ) be the family of all possible packings of n>1 translates of the o-symmetric convex body K o in E d,d\ 3. Moreover, let Pn (K o )}. d(k o, n)=max3 n Vol d (K o ) Vol d (1 n i=1(c i +K o )) : {c 1 +K o,...,c n +K o 4 Then NOTE 197 lim sup n Q. d(k o, n)=d(k o ). Now, we return to the proof of our theorem and look at the packing K={c 1 +K o, c +K o,...,c n +K o } of n translates of the o-symmetric convex body K o points of with centers at the C n ={c 1, c,...,c n } E d. Notice that if c i +K o is (outer) tangent to H(K o ) members of the packing K, then c i +K o 0 (c j +K o ). j ] i, 1 [ j [ n Thus, if m denotes the number of members of K that are touched by exactly H(K o ) members of K, then the (d 1)-dimensional surface volume Svol d 1 (bd(c n +K o )) of the boundary bd(c n +K o ) of the non-convex set C n +K o must satisfy the inequality (5) Svol d 1 (bd(c n +K o )) [ (n m) d 1 Svol d 1 (bdk o ). Finally, the isoperimetric inequality (see, for example, [8]) applied to C n +K o yields (6) Iq(B) [ (Svol d 1(bd(C n +K o ))) d (Vol d (C n +K o )) d 1 =Iq(C n +K o ). Hence, the Lemma, (5) and (6) imply in a straightforward way that (7) 1 d 1 (d(k (d 1)/d o 1 )) Iq(B) 1/d n (d 1)/d [ n m. Iq(K o ) Finally, notice that the convex hull of C n,n>1 must have at least two vertices in E d say, c i and c j. Then it is easy to see that that the number of

7 198 NOTE members of the packing K that are tangent to c i +K o (resp., c j +K o )isat most h(k o ). From this and (7) the Theorem follows in a straightforward way. This completes the proof of the Theorem. 3. PROOF OF COROLLARY 1 Let C n ={c 1, c,...,c n } be an arbitrary set of n>1 points of minimum distance say, in M d,d\ 3. Moreover, let K o be the closed d-dimensional ball of radius 1 centered at the origin o in M d. Obviously, the number t of touching pairs in the packing {c 1 +K o, c +K o,...,c n +K o } of n translates of the convex body K o in E d is equal to the number of times the distance occurs among the points of C n in M d. Hence, if Iq(K o ) [ (d) d, where (d) d is in fact, the isoperimetric quotient of any d-cube in E d, then the Theorem and the inequalities H(K o ) [ 3 d 1,d(K o ) [ 1, 0 [ H(K o ) h(k o ) 1 imply in a straightforward way that t [ 3d 1 d+1 n(d 1)/d, finishing the proof of the first part of Corollary 1. So, we are left with the case when Iq(K o ) > (d) d. Now, let Q be a d-dimensional cube of E d. Then recall the elegant theorem of Ball [] which claims that there is an affine map j: E d Q E d such that Vol d (j(k o ))=Vol d (Q) and Svol d 1 (bd(j(k o ))) [ Svol d 1 (bdq). As a result we get that (8) Iq(j(K o )) [ Iq(Q)=(d) d. Finally, notice that the number of touching pairs in the packing {j(c 1 +K o ), j(c +K o ),..., j(c n +K o )}

8 of n translates of the convex body j(k o ) in E d is precisely t. Thus, (8) and the Theorem and the inequalities H(j(K o )) [ 3 d 1,d(j(K o )) [ 1, 0 [ H(j(K o )) h(j(k o )) 1 imply again in a straightforward way that t [ 3d 1 d+1 n(d 1)/d. This completes the proof of the first part of Corollary 1. Finally, the second part of Corollary 1 can be proved as follows. Observe that it is sufficient to give an upper bound on the number t of touching pairs in the packing {c 1 +Q, c +Q,..., c n +Q} of n translates of the d-dimensional cube Q of E d, where in fact, Q is the closed d-dimensional ball of radius 1 centered at the origin o in M d l.. Using small translations parallel to the edges of Q we can arrange that the convex hull of the points {c 1, c,...,c n } has dimension d in E d and so, it has at least d+1 vertices moreover, the number of touching pairs in the corresponding packing of n translates of Q is still t. Thus, the proof of the Theorem and the claims H(Q)=3 d 1, h(q)= 3 d 1 1 (the second of which is proved in [3]) imply that t [ 3d 1 This completes the proof of Corollary 1. NOTE 199 d+1 n(d 1)/d d+1 (3 d 1 1). ACKNOWLEDGMENTS The author was partially supported by the Hung. Nat. Sci. Found. (OTKA), Grant T09786 and by the Combinatorial Geometry Project of the Research Found. FKFP0151/1999. REFERENCES 1. P. K. Agarwal and J. Pach, Combinatorial Geometry, Wiley, New York, K. M. Ball, Volume ratios and a reverse isoperimetric inequality, J. London Math. Soc. 44 (1991), K. Bezdek and P. Brass, On the k + -neighbour packings and one-sided Hadwiger configurations, preprint, pp. 1 4, Department of Geometry, Eötvös University, January U. Betke, M. Henk, and J. M. Wills, Finite and infinite packings, J. Reine Angew. Math. 453 (1994),

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