Let $P$ be a non-simple polytope and $E$ a face of codimension $k$ with $\dim E \geq 1$. Is it always true that $E$ is an intersection of $k$ facets of $P$? When $P$ is simple, this is always true.
(If it is not true,) it would be grateful if anybody can give me a simplest example.
Let $P\subset\mathbb{R}^n$ be a polytope of dimension $n$. A face $F\in L(P)$ of codimension $k$ is always the intersection of exactly $k$ facets: the affine hulls of facets are hyperplanes in $\mathbb{R}^n$, and the affine hull of $F$ is of dimension $n-k$, therefore the intersection of exactly $k$ hyperplanes (think of the description of $P$ as bounded intersection of closed half-spaces).
The thing is that, in the case of non-simple polytopes this set is not uniquely defined: suppose that $P$ is not simple, and let $v$ be a vertex which is the intersection of $k>n$ facets $F_1,\ldots,F_k$. Then, $v$ is equal to the intersection of any $n$ element subset of $F_1,\ldots,F_k$.