On Ptolemaic metric simplicial complexes

TL;DR

This paper demonstrates that metric simplicial complexes satisfying the Ptolemy inequality are CAT(0) spaces under mild conditions and characterizes those isometric to Euclidean space via metric inversions.

Contribution

It establishes a link between Ptolemy inequality, CAT(0) spaces, and metric inversions in simplicial complexes, providing new characterizations.

Findings

Ptolemy inequality implies CAT(0) under certain conditions

Characterization of Euclidean simplicial complexes via metric inversions

Conditions under which complexes are CAT(0) or Euclidean

Abstract

We show that under certain mild conditions, a metric simplicial complex which satisfies the Ptolemy inequality is a CAT(0) space. Ptolemy's inequality is closely related to inversions of metric spaces. For a large class of metric simplicial complexes, we characterize those which are isometric to Euclidean space in terms of metric inversions.