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arXiv · 2512.23651

Notes on non-separable arrangements of convex bodies

Károly Bezdek·Zsolt Lángi
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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 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 $0<k<d-1$.

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Károly Bezdek, Zsolt Lángi. 2025-12-29. Notes on non-separable arrangements of convex bodies. https://arxiv.org/abs/2512.23651

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