Convex hulls in geodesic spaces of nonpositive curvature

It is an easy exercise for students having their first brush with convex analysis to prove that the convex hull of a compact set in Euclidean space is compact. This fact essentially persists in Hilbert space, still at the level of a basic exercise: the closed convex hull of a compact set is compact. Moving beyond linearity, one finds the class of complete geodesic metric spaces of nonpositive curvature in the CAT(0) sense. Such spaces, called Hadamard spaces, are nonlinear generalizations of Euclidean and Hilbert spaces, which support a well-developed theory of convexity and a variety of applications in different areas of mathematics. In Hadamard spaces, the question "is the closed convex hull of a compact set necessarily compact?" jumps sharply from a basic exercise to a complete unknown, where it has remained opaque for at least three decades after being publicized by Gromov. In this talk we finally put this question to rest: there is a Hadamard space and a finite set within it whose closed convex hull is not compact. Notably, the example was discovered using AI and the techniques for constructing the example rely only on standard tools from metric geometry. We will provide a brief overview of Hadamard spaces and related work on the behavior of convex hulls in such spaces, before proceeding to walk through the construction of this pathological example. We will present two closely related constructions (representing the order in which they were discovered) beginning with a set of finite but large cardinality and improving it to a set of minimal size containing only three points.
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