A finite set with noncompact closed convex hull in a Hadamard space
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.
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BibTeX
@misc{goodwin2026finite,
title = {{A finite set with noncompact closed convex hull in a Hadamard space}},
author = {Ariel Goodwin},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.06455},
eprint = {2609.06455},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.06455},
}
RIS
TY - UNPB
TI - A finite set with noncompact closed convex hull in a Hadamard space
AU - Ariel Goodwin
PY - 2026
DA - 2026/09/29
PB - arXiv
DO - 10.48550/arxiv.2609.06455
UR - https://arxiv.org/abs/2609.06455
AB - 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.
ER -