A carousel property for compact convex sets

2025/12/16 by Yiming Song, Song, Yiming
Mathematics · Computer Science · #Limits and Structures in Graph Theory #Point processes and geometric inequalities #Computational Geometry and Mesh Generation

paper · pdf · doi:10.48550/arxiv.2512.14972

Abstract

We prove that if A₀ and A₁ are compact convex sets contained in a convex n-gon with vertices G₁, …, G_n, and 2 \lceil (n)/(2) \rceil is strictly greater than the number of common supporting lines of A₀ and A₁, then there exist i∈ \0,1\ and j∈ \1, …, n\ such that A_i is in the convex hull of A_1-i and \G₁,…,G_n\∖\G_j\. This result and its proof recover and generalize results of Adaricheva-Bolat and Czédli-Kurusa. We also show that this bound is sharp for all n. Our results yield explicit examples of convex geometries that cannot be represented by translates of convex bodies in the plane.

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