The boundary $\partial S$ of a compact convex set is simply connected if $\dim\partial S\geq2$

Question

I was wondering if anyone has a reference (e.g., a textbook) for the statement in the title: 'The boundary $\partial S$ of a compact convex set $S\subset\mathbb{R}^n$ is simply connected if $\dim\partial S \geq 2$.'

A related question is:

Connected-ness of the boundary of convex sets in $\mathbb R^n$ , $n>1$ , under additional assumptions of the convex set being compact or bounded

where it is shown that the boundary is connected by a homeomorphism of $\partial S$ with the sphere $\mathcal{S}^{n-1}$. To my understanding this also implies that $\partial S$ is simply connected (of course, we also need $\dim\partial S>1$ for simple connectedness). However, I prefer to have a reference since I can't really digress on this point in the paper I am writing.