Combinatorics- Polytopes Question - MathOverflow [closed] most recent 30 from http://mathoverflow.net 2013-05-24T18:52:46Z http://mathoverflow.net/feeds/question/85642 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/85642/combinatorics-polytopes-question Combinatorics- Polytopes Question jason mfash 2012-01-14T08:50:10Z 2012-01-14T09:22:52Z <p>Can someone help me solve the following question please? </p> <p>Let v be a vertex of a d-polytope P such that $0 \in intP$ . Prove that $P^* \cap \{ y \in \mathbb{R}^d \mid\left &lt; y, v\right>=1\ \}$ is a facet of $P^{*}$. </p> <p>The definitions are: $P^*=\{ y\in\mathbb{R}^{d}\mid\left &lt; x, y\right>\leq 1\ \forall x\in P\}$ and a face of P is the empty set, P itself, or an intersection of P with a supporting hyperplane (i.e.- a hyperplane, such that P is located in one of the halfspaces it determines). A facet is a face of maximal degree</p> <p>I tried showing that if there exists a vertex v such that this isn't a facet, then P is a convex hull of a finite set not containing v, which is a contradiction, but without success.</p> <p>HOpe you'll be able to help me</p>