on joint work with P. Prozorov and Y. Teplitskaya

The Steiner minimal tree problem is to find a set (actually a tree) of the smallest length that contains a given set of points A in Euclidean space. A set is called Ahlfors--David regular if for a sufficiently small radius and any point in the set, the length of the set in the corresponding neighborhood is estimated on both sides by constants times the radius. Since a Steiner tree contains a given set, the question makes sense for all balls that do not intersect the given set A. We discuss Ahlfors--David regularity of planar Steiner trees and some milder results in higher dimensions.