On the Homotopy Type of Balanced subsets

2025/12/09 by Bludov, Mikhail V.
Computer Science · Mathematics · #Combinatorics (math.CO) #Computational Geometry and Mesh Generation #FOS: Mathematics #Point processes and geometric inequalities #Topological and Geometric Data Analysis

paper · doi:10.48550/arxiv.2512.08707

openalex publication_date 2025/12/09 · openalex created_date 2025/12/11 · openalex updated_date 2026/07/28

Abstract

For a finite set of points V=\v₁, …, v_m\ in Euclidean space ℝ^d and a point r ∈ ℝ^d, a subset S ⊂ V is called r-balanced if relint(conv(S)) ∩ r ≠ ∅. In the case when r is a point in the relative interior of the whole set conv(V), we prove that the poset of all balanced subsets, excluding the whole set V, is homotopy equivalent to the sphere of dimension m-k-2, where k is the dimension of the affine hull of V.

Citations

• Colourful Carathéodory's Theorem plus a constraint
• Balanced sets and homotopy invariants of covers
• Balanced 2-subsets
• Maximal unbalanced families

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