Notes on non-separable arrangements of convex bodies

2025/12/29 by Bezdek, Károly, Lángi, Zsolt
#FOS: Mathematics #Metric Geometry (math.MG)

paper · doi:10.48550/arxiv.2512.23651

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

Citations

• On the Circle Covering Theorem by A. W. Goodman and R. E. Goodman
• On densities of lattice arrangements intersecting every i-dimensional\n affine subspace
• Inequalities for the lattice width of lattice-free convex sets in the plane

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• On separability in discrete geometry
• Difference between families of weakly and strongly maximal integral lattice-free polytopes
• Separation in Simply Linked Neighbourly 4-Polytopes
• Complements of finite unions of convex sets
• Realization spaces of arrangements of convex bodies