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        Questions tagged [convex-polytopes]

        Convex polytopes are the convex hulls of a finite set of points in Euclidean spaces. They have rich combinatorial, arithmetic, and metrical theory, and are related to toric varieties and to linear programming

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        The stability of optimal transport ground cost matrices

        I would like to better understand the stability of the ground cost matrix $C \in \mathbb{R}^{n \times n}_+$ within the discrete Kantorovich optimal transport problem: $$\mathcal{P}(C) := \arg \min_{P ...
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        181 views

        Formula for volume of a convex polytope

        So I've been searching around the internet for some answers to this, but I currently have a set of linear constraints: $Ax = b, Cx \le d$ for matrices $A \in \mathbb{R}^{n \times m}$, $b\in \mathbb{R}^...
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        235 views

        Convex hull of all rank-$1$ $\{-1, 1\}$-matrices?

        Consider the set $\mathbb{R}^{m \times n}$ of $m \times n$ matrices. I am particularly interested in properties of polytope $P$ defined as a convex hull of all $\{-1,1\}$ matrices of rank $1$, that is,...
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        Is a polytope with vertices on a sphere and all edges of same length already rigid?

        Let's say $P\subset\Bbb R^d$ is some convex polytope with the following two properties: all vertices are on a common sphere. all edges are of the same length. I suspect that such a polytope is ...
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        What is known about the combinatorics of the hyperplane arrangement spanned by cyclic polytopes?

        Let $1\leq d$ be an integer. Consider the $d$-dimensional moment curve $\mu\colon \mathbb R\to \mathbb R^d$ given by $t\mapsto (t,t^2,\dots, t^d)$. Given a finite subset $S\subset \mathbb R$ of ...
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        1answer
        67 views

        Algorithms for projecting a point onto the convex hull spanned by a set of vectors

        Given a set of vectors $V = \{ \mathbf{v}_1, \ldots, \mathbf{v}_n \} \subset \mathbb{R}^d$, I want to project a point $\mathbf{x}_0 \in \mathbb{R}^d$ onto the convex hull $\text{conv}(V)$ of the ...
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        377 views

        Positivity of the coefficients of the Ehrhart polynomial of a cross-polytope

        Question 35996 asks about the Ehrhart polynomial $i_d(n)$ of the standard regular cross-polytope. It can be defined equivalently by $$ \sum_{n\geq 0}i_d(n)x^n = \frac{(1+x)^d}{(1-x)^{d+1}}. $$ It ...
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        125 views

        Is a vertex- and edge-transitive polytope already a uniform polytope?

        I want to consider (convex) polytopes $P=\mathrm{conv}\{p_1,...,p_n\}\subset\Bbb R^d$ which are both, vertex- and edge-transitive (or maybe stronger: 1-flag-transitive). Question: Is every such ...
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        58 views

        Intersection between a line and a n dimensional parallelotope

        Suppose that I have a line in an $n$ dimensional space described by $$ X=A+Bk, \quad \quad X,A,B \in \mathbb{R}^n, k \in \mathbb{R} $$ here $A$ is known and I want to find all the possible vectors $B$ ...
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        Are there half-transitive convex polytopes?

        I only consider convex polytopes, i.e. convex hulls of finitely many points. The (edge-)graph of a polytope $P\subseteq\Bbb R^d$ is the graph consisting of the polytope's vertices, two are adjacent if ...
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        1answer
        76 views

        Sampling uniformly from the vertices of a polytope

        I'm looking for a reference on how to sample uniformly (and preferably efficiently, elegantly, etc.) from the vertices of a polytope. I gather that enumerating vertices is hard. I also note the MO ...
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        78 views

        Name for facet of a cone containing all but one edge

        Let $C \subseteq \mathbb R^n$ be a polyhedral cone, so generated by its edges ($1$-dimensional faces) and $F \subseteq C$ a facet (codimension $1$ face) of it containing every edge except $e$. In ...
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        Counting Zeros Under Unitary Action

        Assume we have two polynomials $f_1$ and $f_2$ with Newton polytopes $A_1 , A_2 \in \mathbb{Z}^2$. Also suppose that coefficients of $f_1$ and $f_2$ are generic. Then we pick a unitary matrix $Q$ and ...
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        Laplace Beltrami eigenvalues on surface of polytopes

        The recently posted arxiv paper Spectrum of the Laplacian on Regular Polyhedra by Evan Greif, Daniel Kaplan, Robert S. Strichartz, and Samuel C. Wiese, collects numerical evidence for conjectured ...
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        73 views

        Existence of a “generic enough” lattice point interior to a lattice triangle

        Let $T$ be a lattice triangle in $\Bbb R^2$ (i.e. the convex hull of three noncolinear points in $\Bbb Z^2$), and assume it has at least one interior lattice point. Is it always possible to find a ...

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