Fix a compact convex subset $C \subset \mathbb{R}^n$ with nonempty interior. For any subspace $S \subset \mathbb{R}^n$, let $P_S$ denote the orthogonal linear projection onto $S$. I'd like to claim that for almost every (in either a measure theory or topological sense) nontrivial subspace $S$ of a given dimension, the image of $C$ under $P_S$ has the following property: for every $y$ in the relative boundary of $P_S(C)$, the fiber $\{x \in C : P_S(x) = y\}$ is a singleton.

In the case when $\dim S = 1$, my claim is equivalent to the statement that almost every linear functional attains its maximum at a single point in $C$. I'm particularly interested in the case when $\dim S = n-1$. Has anyone ever seen anything like this? Note: I believe it is easy to verify the claim when $C$ is strictly convex or polyhedral, so the interesting situation is when $C$ is a general compact convex set.

Thinking about this some more, it is not even obvious that the set of subspaces $S$ of a given dimension with this property (points on the relative boundary of $P_S(C)$ have unique preimages in $C$) is nonempty. Even a basic existence argument for general convex sets would be useful.