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

Let Hd denote the smallest integer n such that for every convex body K in R there is a 0 < λ < 1 such that K is covered by n translates of λK. In [2] the following problem was posed: Is there a 0 < λd < 1 depending on d only with the property that every convex body K in R is covered by Hd translates of λdK? We prove the affirmative answer to the question and hence show that the Gohberg–Markus–Boltyanski–Hadwiger Conjecture (according to which Hd ≤ 2 ) holds if, and only if, a formally stronger version of it holds. 1. Definitions and Results A convex body in R is a compact convex set K with non–empty interior. Its volume is denoted by vol(K). Definition 1.1. For d ≥ 1 let Hd denote the smallest integer n such that for every convex body K in R there is a 0 < λ < 1 such that K is covered by n translates of λK. Furthermore, let Hd denote the smallest integer m such that there is a 0 < λd < 1 with the property that every convex body K in R d is covered by m translates of λdK. Clearly, Hd ≤ Hd. The following question was raised in [2] (Problem 6 in Section 3.2): Is it true that Hd = Hd? We answer the question in the affirmative using a simple topological argument. Theorem 1.2. Hd = Hd. The famous conjecture of Gohberg, Markus, Boltyanski and Hadwiger states that Hd ≤ 2 (and only the cube requires 2 smaller positive homothetic copies to be covered). For more information on the conjecture, refer to [1], [7] and [11]. In view of Theorem 1.2, the conjecture is true if, and only if, the following, formally stronger conjecture holds: Conjecture 1.3 (Strong Gohberg–Markus–Boltyanski–Hadwiger Conjecture). For every d ≥ 1 there is a 0 < λd < 1 such that every convex body K in R is covered by 2 translates of λdK. In Section 2 we prove the Theorem. We note that the proof provides no upper bound on λd in terms of d. In Section 3 we show an upper bound on the number of 1991 Mathematics Subject Classification. 52A35, 52A20, 52C17.