发表机构
Cornell University（康奈尔大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文通过AI生成的基于CAT(1)度量图归纳序列及欧氏锥的构造，解决了Gromov提出的Hadamard空间中有限集闭凸包非紧性问题，并讨论了历史及相关问题。
AI 中文摘要
我们通过一个AI生成的构造证明了Hadamard空间中有限集的闭凸包可以是非紧的。寻找这样一个集合或证明其不存在这一看似简单的问题，由Gromov提出，并被度量几何、泛函分析、不动点理论和几何群论领域的研究者定期重新审视，三十多年来一直未获解答。该构造本身基于构建一个CAT(1)度量图的归纳序列，并对极限空间取欧氏锥。我们讨论了该问题的历史，解释了构造及其与现有技术的关系，并提及了一些相关问题。
英文摘要
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.
Comments24 pages, 5 figures. This version improves the result of the first version (from a finite set to three-point set), alongside several improvements to the overall presentation
