Description
Abstract: One way of relaxing convexity assumptions is to consider weakly convex sets and functions.
In Euclidean spaces, there are various equivalent definitions of weak convexity. One straightforward definition relates weak convexity to strong or spindle convexity: a set $A$ is weakly convex if, for any two points $x$ and $y$ within a distance of at most 2, the spindle with endpoints $x$ and $y$ contains at least one additional point from the set $A$.
The primary focus of this talk is to outline the geometric proof of a fundamental result: within a weakly convex set, there exists a unique shortest path between any pair of points separated by a distance strictly less than two.
This presentation is derived from joint work with G.E. Ivanov and M.S. Lopuchanski, available at arxiv.org/abs/2308.15279.