Notes on non-separable arrangements of convex bodies

TL;DR

This paper advances the understanding of non-separable arrangements of convex bodies by extending minimal covering results to weakly non-separable families, analyzing stability, and exploring maximal and weakly k-impassable families of convex polytopes.

Contribution

It generalizes existing results on minimal coverings from NS-families to weakly non-separable families, including stability analysis and new insights into maximal and weakly k-impassable families.

Findings

Established analogues for weakly non-separable families of convex polytopes.

Derived stability results for these families.

Analyzed maximal weakly non-separable families of cubes and weakly k-impassable families.

Abstract

A problem posed by Erd\H{o}s in 1945 initiated the study of non-separable arrangements of convex bodies. A finite collection of convex bodies in Euclidean $d$-space is called a non-separable family (or NS-family) if every hyperplane intersecting their convex hull also intersects at least one member of the family. Recent work has focused on minimal coverings of NS-families consisting of positive homothetic convex bodies. In this paper, we strengthen these results by establishing their analogues for weakly non-separable families of convex polytopes. We further obtain stability results and analyze maximal weakly non-separable families of cubes. As an additional extension, we also examine weakly $k$-impassable families of convex $d$-polytopes for $0<k<d-1$.