# Tagged Questions

In elementary geometry, a polytope is a geometric object with flat sides, and may exist in any general number of dimensions n as an n-dimensional polytope or n-polytope. (Def: http://en.m.wikipedia.org/wiki/Polytope)

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### Nichtnegativstellensatz the same as Handelman's Theorem?

Wikipedia on "Handelman's theorem: If $K$ is a compact polytope in Euclidean $d$-space, defined by linear inequalities $g_i ≥ 0$, and if $f$ is a polynomial in $d$ variables that is positive on $K$, ...
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### Handelman's Theorem on Nonnegative polynomial in Compact Polytope?

Background: Representing polynomials by positive linear functions on compact convex polyhedra by David Handelman Consider a polynomial $f\in K[x_1,\ldots,x_n]$ in a compact polytope, related ...
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### Most efficient way of transforming from V-representation to H-representation

What is an efficient way to transform from the v-representation of a convex hull (in terms of vertices) to its h-representation ($Ax \leq b$)?
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### What are the combinatorial types of the facets of a MTBF?

This question is about Main Tridiagonal Birkhoff Faces (or MTBFs), defined in this question. (One goal of this question - to be addressed in a corollary question - is to establish that MTBFs ...
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### Convex hull of union of polytopes in halfspace representation

Suppose I have two polytopes in $\mathbb{R}^n$ given in H-representation as $P_1: \{x | H_1 x\leq b_1 \}$ $P_2: \{x | H_2 x\leq b_2 \}$ My question is, if it is possible to efficiently (i.e., avoid ...
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### convex polytopes where every vertex pairwise shares a facet

for arbitrary dimension, what are the convex polytopes such that all vertices share a facet of some dimension, which is not the top facet (the entire polytope), with all other vertices? One example is ...
For any $A \in \{0,1\}^{m \times n}$ and $r \in \mathbb{R}^m_{>0}$ consider the set $S := \{x \in \mathbb{R}^n_{\geq 0} \ | \ A x = r\}$ of non-negative solutions to the linear system $Ax - r = 0$. ...
I have recently learned the orbit stabilizer theorem, and have encountered unexpected results pertaining to the rotations of a tesseract; I am curious if there is any intuition for this. A $4$-Cube ...