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Let $P$ and $V$ be, respectively, a bounded full-dimensional polytope and an $m$-dimensional affine subspace of $\mathbb R^n$, and assume $P \cap V$ is non-empty.

Q1. Could you provide me with a reference where the following (or even better a generalization of it) is explicitly stated and proved?

Lemma. There exists a face $F$ of $P$ such that $\dim(F) \le n-m$ and $F \cap V \ne \emptyset$.

This is certainly well-known for everyone familiar with polyhedral theory (I'm stressing it due to the tone of certain comments below), but I wouldn't say that it is trivial (whatever this may mean), and I need it as an intermediate step in the proof of a certain result in zero-sum theory. Yet, I would like not to include it in my paper, since otherwise I feel that, in the interest of my two readers, I should also introduce some lexicon and so on. That's why I'm looking for a reference.

In particular, I gave a look at A. Brøndsted's An Introduction to Convex Polytopes and G. M. Ziegler's Lectures on Polytopes, but I don't seem to find it. Thank you in advance for any help.

Q2 (added later). Is there any "non-trivial" generalization of the lemma to a larger class of convex sets than the one of polytopes or to something different from an affine subspace?

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    $\begingroup$ By connection, $V$ has to meet the boundary of $P$, which is a union of faces of dimension less than $n$. Then apply induction on $n$. $\endgroup$ Apr 5, 2014 at 8:18
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    $\begingroup$ I was just about to write the same that Pietro did. This does not look like a statement whose proof should be proudly announced in a paper/book. $\endgroup$ Apr 5, 2014 at 8:33
  • $\begingroup$ I don't really understand the tone of these comments. Books and papers are full of simple statements with a one-line proof which turn out to be pretty useful for some specific purposes. Are you claiming that there's something wrong with that?! $\endgroup$ Apr 5, 2014 at 15:22
  • $\begingroup$ @Pietro Majer. I agree that the standing assumptions and $\dim(V)\ne 0$ imply $V\cap{\rm bd}(P)\ne\emptyset$, in view of the following: "If $C$ and $S$ are, resp., a compact and a connected unbounded subset of $\mathbb R^n$ s.t. $C\cap S\ne\emptyset$, then $S$ meets ${\rm bd}(P)$" (which in turn is an easy consequence of Proposition 3, Ch. I, Sect. I.1 in Bourbaki's General Topology: Part 1). I also agree that the bd of a polytope is the union of its proper faces. Yet, I think this is not enough for your induction to work. That said, I'm really looking for a reference, not for a proof. $\endgroup$ Apr 9, 2014 at 17:49

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