Given 2 homotopic curves in a topological space, there are several ways to measure similarity between the curves, including Hausdorff distance and Fréchet distance. In this paper, we examine a different measure of similarity which considers the family of curves represented in the homotopy between the curves, and measures the longest such curve, known as the height of the homotopy. In other word...

We prove that there exists a norm in the plane under which no n-point set determines more than O(n log n log logn) unit distances. Actually, most norms have this property, in the sense that their complement is a meager set in the metric space of all norms (with the metric given by the Hausdorff distance of the unit balls).

A graph is said to be distance-hereditary if the distance function in every connected induced subgraph same as itself. We prove that ordinary Weisfeiler–Leman algorithm tests isomorphism of any two graphs one them distance-hereditary; more precisely, dimension class finite equal 2. The previously best known upper bound for was 7.