Structure of average distance minimizers in general dimensions

Abstract

For a fixed, compactly supported probability measure $\mu$ on the $d$-dimensional space $\mathbb{R}^d$, we consider the problem of minimizing the $p^{\mathrm{th}}$-power average distance functional over all compact, connected $\Sigma \subseteq \mathbb{R}^d$ with Hausdorff 1-measure $\mathcal{H}^1(\Sigma) \leq l$. 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 \textit{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 $p > \frac{1}{2}(3 + \sqrt{5}) \approx 2.618$, the first such result that includes the case when $d > 2$.