If we project a convex set on a lower-dimensional subspace, take a face of this projection, and pullback this face to the original set, what do we get?

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

Does projecting a convex set preserve faces of that set in some way?

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 $X$?

If we project a convex set on a lower-dimensional subspace, take a face of this projection, and pullback this face to the original set, what do we get?

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

If we project a convex set on a lower-dimensional subspace, take a face of this projection, and pullback this face to the original set, what do we get?

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

Does projecting a convex set preserve faces of that set in some way?