On the Constant of Homothety for Covering a Convex Set with Its Smaller Copies

ON THE CON ST ANT OF HOMO THETY F OR COVERING A CONVEX SET WITH ITS SMALLER COPIES M ´ AR TON NASZ ´ ODI ∗ Abstract. Let H d denote the smallest integ er n s uc h that f or ev ery conv ex bo dy K in R d there is a 0 < λ < 1 such that K is co vered by n translates of λK . In [2] the following problem wa s p osed: Is there a 0 < λ d < 1 depending on d only with the property that every conv ex b o dy K in R d is cov er ed by H d translates of λ d K ? W e prov e the affirmative answe r to the question and hence sho w that the Gohberg–Markus–Bolt yan ski–Hadwiger Conjecture (according to which H d ≤ 2 d ) holds if, and only if, a form al ly stronger v ersion of it holds. 1. Definitions and Resul ts A c onvex b o dy in R d is a co mpact co nv ex set K with non–empty interior. Its volume is denoted by vol( K ). Definition 1.1. F or d ≥ 1 let H d denote the smallest integer n such tha t for ev ery conv ex b o dy K in R d there is a 0 < λ < 1 such that K is covered by n translates of λK . F urther mo re, let H d denote the s mallest integer m such that there is a 0 < λ d < 1 with the prop er t y that ev ery conv ex b o dy K in R d is c overed by m translates o f λ d K . Clearly , H d ≤ H d . The following question w as r aised in [2] (Pr oblem 6 in Section 3.2): Is it tru e that H d = H d ? W e answer the ques tion in the a ffirmative using a simple topo lo gical ar gument. Theorem 1.2. H d = H d . The famous conjecture of Gohber g, Mark us, Bolty a nski and Hadwiger states that H d ≤ 2 d (and only the cube r equires 2 d smaller pos itive homothetic copies to be cov ered). F or more information on the c o njecture, refer to [1], [7] and [11]. In view of Theo rem 1.2, the conjecture is true if, and only if, the following, formally stronger conjecture holds: Conjecture 1.3 (Strong Go h b erg– Markus–B o lty a nski–Hadwiger Conjecture) . F or every d ≥ 1 t her e is a 0 < λ d < 1 s uch t hat every c onvex b o dy K in R d is c over e d by 2 d tr anslates of λ d K . In Section 2 we pr ov e the Theorem. W e note that the pro o f provides no upp er bo und on λ d in terms of d . In Section 3 we show a n upper b ound on the nu mber of 1991 Mathematics Subje c t Classific ation. 52A35, 52A20, 52C17. Key wor ds and phr ases. illumination, Bolt yanski–Had wiger Conjecture, con vex sets. ∗ Pa rtially supp orted by a Postdoctoral F ellowship of the Pa cific Institute for the Mathematical Sciences. 1 2 M. NASZ ´ ODI translates of λK required to co ver K , improving a result o f Ja nu szewsk i and La ssak [5]. 2. Proof of Theorem 1. 2 W e define the following function on the set of convex b o dies: λ ( K ) := inf { λ > 0 : K is covered by H d translates o f λK } . By [8], H d is finite for every d , so λ ( . ) is well defined. Remark 2. 1. Clearly , λ ( . ) is affine inv aria n t; tha t is, if T is a n inv ertible affine transformatio n of R d then λ ( K ) = λ ( T K ). Moreover, 0 < λ ( K ) < 1. W e recall the definition o f the (multiplicativ e) Banach–Mazur distanc e of tw o conv ex b o dies L and K in R d : (2.1) d BM ( L, K ) = inf { λ > 0 : L − a ⊆ T ( K − b ) ⊆ λ ( L − a ) for some a, b ∈ R , T ∈ GL ( R d ) } The follo wing pro p o s ition sta tes that λ ( . ) is uppe r semi–co nt inuous. Similar statements have b een prov ed b efore, cf. Lemma 2. in [3]. Prop ositio n 2. 2 . F or every c onvex b o dy K and ε > 0 ther e is a δ > 0 with the pr op erty that for any c onvex b o dy L , if d BM ( L, K ) < 1 + δ then λ ( L ) < λ ( K ) + ε . Pr o of. Let λ := λ ( K ) + ε 2 . Then there is a set Λ ⊂ R d with card Λ ≤ H d such that K ⊆ Λ + λK . Now, let δ > 0 b e such that (2.2) 1 + δ < λ + ε 2 λ Assume that d BM ( L, K ) < 1 + δ ; that is, (2.3) L − a ⊆ ¯ K ⊆ (1 + δ )( L − a ) , where ¯ K is an a ffine ima g e (under an in vertible affine transfor mation) of K . Clea rly , we may as s ume that ¯ K = K . It follows that L − a ⊆ Λ + (1 + δ ) λ ( L − a ), and hence, λ ( L ) ≤ (1 + δ ) λ < λ ( K ) + ε .  Let K d a denote the set of affine equiv alence cla sses of convex bo dies in R d equipp e d with th e topo lo gy induced b y the metric d BM . In [6] it is shown that K d a is a compact space. (Note that Macb eath use s a different metric on K d a how ever, that metric induces the same top olo gy as d BM , c f. [4].) It follo ws from Remark 2.1 and Pr op osition 2.2 tha t λ ( . ) is an upper semi– contin uo us function on a compact space . Hence, it a tta ins its maximum, which (b y Remark 2.1) is less than one. This prov es The o rem 1 .2. ON THE CONST ANT OF HOMOTHETY 3 3. Quantit a tive Resul ts Januszewski and Lassak [5] proved that for every k + l > d d , any conv ex bo dy K ⊂ R d is cov ered by k translates of λK and l translates of − λK , where λ = 1 − 1 ( d +1) d d . The following argument shows that o ne may obtain a b etter es timate on the n umber of tra nslates of λK requir ed to cover K , using r esults o f Rogers [8], Rogers a nd Shephard [9], and Roger s and Zhong [10]. Let K , L b e co n vex b o dies in R d . L e t N ( K , L ) deno te the c overing n umb er of K and L ; that is, the smallest n umber of translates of L required to cov er K . In [10] it is s hown that N ( K , L ) ≤ vol ( K − L ) vol ( L ) Θ( L ) , where Θ ( L ) is the covering dens ity of L . By [8 ], Θ ( L ) ≤ d log d + log log d + 5 d for every co nv ex bo dy L in R d . It follows that for a n y 0 < λ < 1 w e hav e N ( K , λK ) ≤ λ − d vol ( K − K ) vol K ( d lo g d + log log d + 5 d ) ≤ λ − d  2 d d  ( d lo g d + log log d + 5 d ) (3.1) The la st inequality follows from the Rog ers–Shephar d Inequa lit y[9]. Similarily , N ( K , − λK ) ≤ λ − d vol ( K + K ) vol K ( d lo g d + log log d + 5 d ) = λ − d 2 d ( d lo g d + log log d + 5 d ) (3.2) By substituting λ = 1 2 int o (3.1) and (3.2), we obtain the following: Remark 3.1 . The num b er of trans lates of 1 2 K that cover K is o f order not greater than 8 d √ d lo g d ; and the num ber of translates of − 1 2 K that cover K is o f order no t greater than 4 d d lo g d . Definition 3.2. Le t 0 < λ < 1, and d ≥ 1. W e denote by H d ( λ ) the smallest int eger n such that every co n vex bo dy K in R d is covered by n translates of λK . It follows from Remar k 3.1 that H d ( 1 2 ) is finite for every d . A natural s tr ength- ening of the question we discussed in this note is the following: Question 3.3. Is there a universal consta nt 0 < λ < 1 such that for ev ery dimen- sion d , H d is equal to H d ( λ )? References [1] Bezdek, K. The il lumination co nje ctur e and its extensions. Period. Math. Hungar. 53 (2006), no. 1-2, 59–69. [2] Br ass, P .; M oser, W.; P ac h, J. R ese ar ch pr oblems in discr e te ge ometry. Spri nger, New Y ork, 2005. xii+499 pp. ISBN: 978-0387-23815-8; 0-387-23815-8 [3] Boltjanski, V. G. ; Soltan, P . S. A solution of H adwiger’s co vering pr oblem for zonoids. Comb inatorica 12 (1992), no. 4, 381–388. [4] Gr ¨ un baum, B. Me asur es of symmetry for co nvex sets. 1963 Pro c. Sympos. Pure Math., V ol. VII pp. 233–270 Amer. Math. Soc., Provide nce, R.I. [5] Januszewski, J.; Lassak, M. Covering a c onvex b o dy by its ne gative homo thetic c opies. Pac ific J. Math. 197 (2001), no. 1, 43–51. 4 M. NASZ ´ ODI [6] M acbeath, A . M. A c omp actness the or em for affine e quivalenc e-classes of c onvex r e gions. Canadian J. Math. 3 , (1951). 54–61. [7] M artini, H., Soltan, V. , Combinatorial pr oblems on the il lumination of co nvex b o dies , Ae- quitiones Math. 57 (1999), 121–152. [8] Rogers, C. A. A note on c overings Mathematik a 4 (1957), 1–6. [9] Rogers, C. A. ; Shepha rd, G. C., The diffe r ence b o dy of a c onvex b o dy , Arch. Math. 8 (1957), 220–233. [10] Rogers, C. A.; Zong, C. Covering c onvex b o dies by t ra nslates of co nvex b o dies. Mathematik a 44 (1997), no. 1, 215–218. [11] Szab´ o, L. Re c ent r esults on il lumination pr oblems , Intuitiv e geometry (Budap est, 1995), 207– 221, Bolyai So c. Math. Stud., 6 , J´ anos B´ olyai Math. Soc., B udap est, 1997. M ´ ar ton Na sz ´ odi, Dept. of Ma th. and St a ts., 632 Central Academic Building, Uni- versity of Alber t a, Edmonton, Ab , Cana da T6G 2G1 E-mail addr ess : mnaszodi@math.u alberta.ca