Non-separable arrangements of convex bodies revisited

Abstract

A problem posed by Erdős 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 stability analogues and analyze weakly non-separable families of convex polytopes. We also examine weakly $k$-impassable families of convex $d$-polytopes for $0<k<d-1$ and study tightness of NS-lattice arrangements.

arXiv comment19 pages, 3 figures