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

TL;DR

This paper proves that for every dimension, a fixed homothety ratio exists such that any convex body can be covered by a finite number of its smaller translates, confirming a long-standing conjecture in discrete geometry.

Contribution

It establishes the existence of a universal homothety ratio depending only on dimension that covers convex bodies with a finite number of smaller copies, affirming a key conjecture.

Findings

Confirmed the existence of a dimension-dependent homothety ratio.

Showed the equivalence of the Gohberg–Markus–Boltyanski–Hadwiger Conjecture to a stronger version.

Provided a positive answer to a longstanding open problem in discrete geometry.

Abstract

Let $H_d$ denote the smallest integer $n$ such that for every convex body $K$ in $\Re^d$ there is a $0<\lambda < 1$ such that $K$ is covered by $n$ translates of $\lambda K$. In the book \emph{Research problems in discrete geometry.} by Brass, Moser and Pach, the following problem was posed: Is there a $0<\lambda_d<1$ depending on $d$ only with the property that every convex body $K$ in $\Re^d$ is covered by $H_d$ translates of $\lambda_d K$? We prove the affirmative answer to the question and hence show that the Gohberg--Markus--Boltyanski--Hadwiger Conjecture (according to which $H_d\leq 2^d$) holds if, and only if, a formally stronger version of it holds.