Homothetic covering of convex hulls of compact convex sets

2021/04/18 by Senlin Wu, Keke Zhang, Wu, Senlin +3
Mathematics · #52A10 #52A15 #52A20 #52C17 #Combinatorics (math.CO) #FOS: Mathematics #Metric Geometry (math.MG) #Point processes and geometric inequalities

paper · pdf · doi:10.48550/arxiv.2104.08868

openalex publication_date 2021/04/18 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let K be a compact convex set and m be a positive integer. The covering functional of K with respect to m is the smallest λ∈[0,1] such that K can be covered by m translates of λK. Estimations of the covering functionals of convex hulls of two or more compact convex sets are presented. It is proved that, if a three-dimensional convex body K is the convex hull of two compact convex sets having no interior points, then the least number c(K) of smaller homothetic copies of K needed to cover K is not greater than 8 and c(K)=8 if and only if K is a parallelepiped.

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