Discussion https://mathoverflow.net/questions/6627/convex-hull-in-cat0 indicates the convex hull of a finite set can fail to be closed in a complete Hadamard space. Hence the following question should have a negative answer:
Suppose K is a compact convex set in the complete Hadamard (or $\operatorname{CAT}(0)$) space $X$, and suppose $K$ is the closed convex hull of the finite point set $\{x_1,x_2,\dots,x_N\}$. Suppose $x$ is in $X$ and $C$ is the cone over $K$ with respect to $x$, i.e. $C$ is the union of all the geodesics connecting $x$ to $K$. Must $C$ be convex?
If not, which value of $N$ yields a minimal counterexample? Is it $N=3$? Can someone describe a simple counterexample?