Let $X\subseteq\mathbb{R}^n$ be a convex set. Let $\pi{:}\ X\to\mathbb{R}^m$ be a linear map, with $m<n$ (for example, a projection). Let $\pi^{-1}(y)=\{x\in X\mid\pi(x)=y\}$ denote the inverse of $\pi$. If $F$ is a face of $\pi(X)$, what is the set $\pi^{-1}(F)$? Is it a face of $X$?