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Encountered the following statement while reading a paper where it was stated without proof - am wondering why its true.

Suppose $P$ is a polytope, $M$ is a convex subset of $P$. Define $f(M)$ to be the minimal face of $P$ which contains $M$. Then there is a point in $M$ which is in the relative interior of $f(M)$.

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    @atricks: Because the exterior of a face is formed by smaller faces, if $x$ were to lie in the exterior of $f(M)$, it would also lie in a smaller face, so $f(M)$ wouldn't be minimal.2012-07-14

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