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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.

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I asked this question several years ago, and I recently found the answer in a paper by Ewald, Larman, and Rogers called "The directions of the line segments and of the r-dimensional balls on the boundary of a convex body in Euclidean space". This was published in 1970 in Mathematika, and the main result in the paper is:

Theorem 1. If $K$ is a convex body in $\mathbb{R}^n$, the set $S$, of end-points of the vectors drawn from the origin in the directions of the line segments lying on the surface of $K$, is a set of $\sigma$-finite $(n-2)$-dimensional Hausdorff measure on the $(n-1)$-dimensional surface of the unit ball.

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