# Combinatorics- Polytopes Question [closed]

Can someone help me solve the following question please?

Let v be a vertex of a d-polytope P such that $0 \in intP$ . Prove that $P^* \cap \{ y \in \mathbb{R}^d \mid\left < y, v\right>=1\ \}$ is a facet of $P^{*}$.

The definitions are: $P^*=\{ y\in\mathbb{R}^{d}\mid\left < x, y\right>\leq 1\ \forall x\in P\}$ 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

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.

HOpe you'll be able to help me

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Is this an exercise? What's the source? –  Yemon Choi Jan 14 '12 at 9:13
I have no idea what is the source... I can't find this exercise in any textbook related to discrete geometry... Our teacher gave this exercise in class and I have no idea how to prove it... –  jason mfash Jan 14 '12 at 9:42
This is an exercise, and this unsuitable for MO. –  matthias beck Jan 14 '12 at 17:37
If $v\in P$, then $P^* \cap${$y \in \mathbb{R}^d \mid\left < y, v\right>=1$}=$P^*$ –  Buschi Sergio Jan 14 '12 at 19:18
It is a help forum, but not for exercises. Please see the FAQ. –  S. Carnahan Jan 15 '12 at 13:02
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## closed as too localized by Benjamin Steinberg, Igor Pak, Bill Johnson, Yemon Choi, S. Carnahan♦Jan 15 '12 at 13:02

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