Year: 2026
Type: preprint
Abstract: We show that there is a Hadamard space in which the closed convex hull of a set containing merely three points is noncompact, by providing an AI-generated construction. The seemingly innocuous problem of finding a compact or finite set with this property, or showing that none exists, was recorded by Gromov and periodically revisited by researchers in metric geometry, functional analysis, fixed-point theory, and geometric group theory. Nevertheless, it 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, the main ideas and details of the construction, and its relationship with existing techniques.
Source: arXiv (Cornell University)
Author: Goodwin, Ariel
Institution: Cornell University
Language: English
Subfield: Applied Mathematics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7211991250