A finite set with noncompact closed convex hull in a Hadamard space

RESUMO

We show that the closed convex hull of a finite set in a Hadamard space can be noncompact by providing an AI-generated construction. The seemingly innocuous problem of finding such a set or showing that none exists, recorded by Gromov and periodically revisited by researchers in metric geometry, functional analysis, fixed-point theory, and geometric group theory, remained unanswered for over three decades. The construction itself is based on building an inductive sequence of CAT(1) metric graphs and taking the Euclidean cone over the limiting space. We discuss the history of this problem, explain the construction and its relationship with existing techniques, and mention some related questions.