On k-convex hulls

2024/11/21 by Davide Ravasini, Ravasini, Davide
Computer Science · Mathematics · #52A20 #52A23 #Advanced Topology and Set Theory #FOS: Mathematics #Functional Analysis (math.FA) #Metric Geometry (math.MG) #Optimization and Variational Analysis #Point processes and geometric inequalities

paper · pdf · doi:10.48550/arxiv.2411.14195

openalex publication_date 2024/11/21 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

For every integer k≥ 2 and every R>1 one can find a dimension n and construct a symmetric convex body K⊂ℝⁿ with diam Q_k-1(K)≥ R\cdotdiam Q_k(K), where Q_k(K) denotes the k-convex hull of K. The purpose of this short note is to show that this result due to E. Kopecká is impossible to obtain if one additionally requires that all isometric images of K satisfy the same inequality. To this end, we introduce the dual construction to the k-convex hull of K, which we call the k-cross approximation of K. We also prove an infinite-dimensional version of the main result that holds in general Hilbert spaces.

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