Homotheties and Coverings by Convex Sets

2023/02/17 by Lawrence, Jim
#52A20 #52A40 #FOS: Mathematics #Metric Geometry (math.MG)

paper · doi:10.48550/arxiv.2302.09015

Abstract

It is shown that, for any function g that is weakly increasing on compact convex sets and has the property that if λ≥ 0 and K^′ is a translate of λK then g(K^′) =λg(K), then for any covering \bigcup_i X_i⊇ X of a compact convex set X by finitely many compact convex sets X_i, the inequality g(X) ≤ ∑_i g(X_i) holds. Among the consequences is an affirmative solution of Bang's Plank Problem.

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