Metric Properties of Conflict Sets

In this paper we show that the tangent cone of a conflict set in $R^n$ is a linear affine cone over a conflict set of smaller dimension and has dimension $n-1$. Moreover we give an example where the conflict sets is not normally embedded and not locally bi-Lipschitz equivalent to the corresponding tangent cone.

💡 Research Summary

The paper investigates the geometric and metric structure of conflict sets in Euclidean space ℝⁿ. A conflict set C is defined as the collection of points that are equidistant to at least two of a finite family of closed subsets A₁,…,A_k ⊂ ℝⁿ. While such sets naturally arise in nearest‑neighbor problems, clustering, and robot motion planning, most prior work has focused on their topological aspects or on smooth cases. This study aims to understand the local tangent geometry and the embedding properties of C in the general, possibly non‑smooth situation.

Main Results

1. Tangent Cone Structure
   For any point p ∈ C, the authors consider the scaled sets λ·(C−p) (λ → 0⁺) and take the Hausdorff limit, denoted T_pC. They prove that T_pC is always a linear affine cone of dimension n‑1. Moreover, T_pC can be described as a cone over another conflict set of dimension n‑1: there exists a (n‑1)-dimensional linear subspace L and a normal vector n̂ such that
   \