Convexity is a local property in $CAT(\kappa)$ spaces

TL;DR

This paper proves that in $CAT()$ spaces, connected, closed, and locally convex subsets are globally convex, given certain diameter conditions, extending the understanding of convexity in these geometric spaces.

Contribution

It establishes that local convexity implies global convexity in $CAT()$ spaces under specific diameter constraints, a new insight in metric geometry.

Findings

Connected, closed, locally convex subsets are convex in $CAT()$ spaces.

Additional diameter condition is necessary when  > 0.

Results extend convexity properties known in Euclidean and hyperbolic spaces.

Abstract

In this note we show that a connected, closed and locally convex subset (with an extra assumption on the diameter with respect to the induced length metric if $\kappa>0$) of a $CAT(\kappa)$ space is convex.