Post Closed as "too localized" by Benjamin Steinberg, Igor Pak, Bill Johnson, Yemon Choi, S. Carnahan
2 TeX

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: $P^*={ P^*=\{ y\in\mathbb{R}^{d}\mid\left < x, y\right>\leq 1\ \forall x\in PP\}$ 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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# Combinatorics- Polytopes Question

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