Dejan Slepcev, Carnegie Mellon University

Abstract: The average distance problem asks to find a good way to approximate a high-dimensional object, represented as a measure, by a one-dimensional object. We will discuss two variants of the problem: one where the one-dimensional object is a measure with connected one-dimensional support and one where it is an embedded curve. We will discuss the basic properties and the regularity results of the minimizing objects. We will proceed to consider some functionals that model optimal transport networks and highlight some of their properties. The talk is based on joint works with Xin-Yan Lu and Slav Kirov.