<ruby id="d9npn"></ruby>

      <sub id="d9npn"><progress id="d9npn"></progress></sub>

      <nobr id="d9npn"></nobr>

      <rp id="d9npn"><big id="d9npn"><th id="d9npn"></th></big></rp>

      <th id="d9npn"><meter id="d9npn"></meter></th>

      Stack Exchange Network

      Stack Exchange network consists of 175 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.

      Visit Stack Exchange

      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

      0
      votes
      0answers
      12 views

      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 ...
      5
      votes
      2answers
      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}^...
      16
      votes
      1answer
      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,...
      1
      vote
      1answer
      75 views

      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 ...
      1
      vote
      0answers
      61 views

      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 ...
      4
      votes
      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 ...
      11
      votes
      2answers
      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 ...
      1
      vote
      2answers
      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 ...
      0
      votes
      1answer
      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 $B$ ...
      1
      vote
      0answers
      68 views

      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 ...
      4
      votes
      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 ...
      4
      votes
      0answers
      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 ...
      1
      vote
      0answers
      34 views

      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 ...
      5
      votes
      0answers
      73 views

      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 ...
      2
      votes
      0answers
      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 ...

      15 30 50 per page
      特码生肖图
      <ruby id="d9npn"></ruby>

          <sub id="d9npn"><progress id="d9npn"></progress></sub>

          <nobr id="d9npn"></nobr>

          <rp id="d9npn"><big id="d9npn"><th id="d9npn"></th></big></rp>

          <th id="d9npn"><meter id="d9npn"></meter></th>

          <ruby id="d9npn"></ruby>

              <sub id="d9npn"><progress id="d9npn"></progress></sub>

              <nobr id="d9npn"></nobr>

              <rp id="d9npn"><big id="d9npn"><th id="d9npn"></th></big></rp>

              <th id="d9npn"><meter id="d9npn"></meter></th>