Combinatorial local convexity implies convexity in finite dimensional CAT(0) cubed complexes

TL;DR

This paper proves that in finite dimensional CAT(0) cubed complexes, a connected subcomplex is convex if and only if its links are full subcomplexes of the ambient complex's links, establishing a key convexity criterion.

Contribution

It provides a proof of a well-known theorem linking local combinatorial conditions to convexity in CAT(0) cubed complexes.

Findings

Convexity characterized by local link conditions.

Full subcomplex condition is necessary and sufficient for convexity.

Clarifies the relationship between local combinatorics and global convexity.

Abstract

We give a proof to the following theorem, which is well-known among experts: A connected subcomplex $W$ of a finite dimensional CAT(0) cubed complex $X$ is convex if and only if Lk$(v, W)$ is a full subcomplex of Lk$(v, X)$ for every vertex $v$ of $W$.