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

2009/01/17 by Márton Naszódi, Marton Naszodi, Naszodi, Marton
Mathematics · #52A20 #52A35 #52C17 #Combinatorics (math.CO) #FOS: Mathematics #Metric Geometry (math.MG) #Point processes and geometric inequalities #math.CO #math.MG #msc:52A20 #msc:52A35 #msc:52C17

paper · pdf · doi:10.48550/arxiv.0901.2652

4 pages, no figures

arxiv created 2009/01/17 · openalex publication_date 2009/01/17 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let H_d denote the smallest integer n such that for every convex body K in \Re^d there is a 0<λ< 1 such that K is covered by n translates of λK. In the book Research problems in discrete geometry. by Brass, Moser and Pach, the following problem was posed: Is there a 0<λ_d<1 depending on d only with the property that every convex body K in \Re^d is covered by H_d translates of λ_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≤ 2^d) holds if, and only if, a formally stronger version of it holds.

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