Let $K$ be a locally-finite simplicial complex; that is there is no vertex in $K$'s vertex set which belongs to infinitely many simplices in $K$. As usual, let $|K|$ denote $K$'s geometric realization. In Proposition 4.2.16 (2) of "Geometric Aspects of General Topology" by Katsuro Sakai it is stated that $K$ is locally finite if and only if it is metrizable (and a proof sketch is given).
My question is, suppose additionally that $K$ is a finite set so that $|K|$ is contained in $\mathbb{R}^{2\#K+1}$; then can we construct a metric $d$ on $|K|$ inducing $|K|$'s topology by simply restricting the Euclidean metric to $|K|$? In particular, simplices in $|K|$ are then geodesically convex under $d$ no?
NB, I guess if $K$ is countably infinite then by we can simply assume that $|K|$ is a subset of $\mathbb{R}^{\mathbb{N}}$ with its usual Fréchet metric?