Structure of average distance minimizers in general dimensions

2025/03/29 by Lucas O'Brien, O'Brien, Lucas, Kobayashi, Forest +2 · 1 citation
Mathematics · #FOS: Mathematics #Limits and Structures in Graph Theory #Mathematical Approximation and Integration #Nonlinear Partial Differential Equations #Optimization and Control (math.OC) #Probability (math.PR)

paper · pdf · doi:10.48550/arxiv.2503.23256

openalex publication_date 2025/03/29 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

For a fixed, compactly supported probability measure μ on the d-dimensional space ℝ^d, we consider the problem of minimizing the p^th-power average distance functional over all compact, connected Σ⊆ ℝ^d with Hausdorff 1-measure H¹(Σ) ≤ l. This problem, known as the average distance problem, was first studied by Buttazzo, Oudet, and Stepanov in 2002, and has undergone a considerable amount of research since. We will provide a novel approach to studying this problem by analyzing it using the so-called barycentre field considered previously by Hayase and two of the authors. This allows us to provide a complete topological description of minimizers in arbitrary dimensions when p = 2 and p > (1)/(2)(3 + √(5)) ≈ 2.618, the first such result that includes the case when d > 2.

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