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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About properties of polynomials with common interlacing
Say $\{a_1,a_2,..,a_n\}$ and $\{b_1,b_2,...,b_n\}$ be the real roots of two monic polynomials of degree $n$ which have a common interlacing. (say I have arranged the roots in increasing order)
Can ...
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Relation between non-integral polytopes, integrally closed polytopes and polynomial Erhart quasi-polynomials
By lattice points, I will always mean points in $\mathbb{Z}^n$ and all polytopes here are convex rational polytopes.
If $P$ is an integral polytope, the counting function for the number of lattice ...
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Measure of intersection of polyhedral cone with unit sphere
Let $C$ be a pointed polyhedral cone in $\mathbb{R}^n$ and let $S^{n-1}$ denote the unit sphere in $\mathbb{R}^n$. Given a description of the supporting hyperplanes of $C$ is there an algorithm for ...
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Convex hull of the orbit of a matrix under permutations
Let $P$ be a generic permutation matrix on $\mathbb{R}^n$. For any vector $x \in \mathbb{R}^n$, the convex hull of the set $\{ Px : \; \text{$P$ is a permutation matrix}\}$ is the set of vectors ...
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Regularity of simplices
A triangle is regular, provided it is equilateral, or, also, equiangular. How these conditions generalize to characterizations of regularity of simplices?
In particular, it turns out that
a ...
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Cone over the Join of two topological spaces
Suppose $X$ and $Y$ are topological spaces. Let's define the join $X\ast Y$ as the quotient space $X\times Y\times [0,1]/\sim$, where $\sim$ is the equivalence relation generated by $(x,y,0)\sim(x,y',...
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Does an $n$ dimensional subspace intersect the $n$-facets of the unit cube?
Since my intuition for high dimensional geometry is not always right:
Consider the unit cube in $\mathbb{R}^m$ and for $n\leq m$ denote by $F^n$ the union of the $n$-facets. For what numbers of $m$ ...
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Wasserstein distance, convex polytopes and extreme points
Let us consider convex polytopes with $K$ extreme points in $\mathbb{R}^d$. Let $\mathbf{P}$ be such polytope, and let $\text{ext}(\mathbf{P})$ denote its set of extreme points, so that $\mathbf{P} = \...
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Algorithm for Finding the Center of an Optimal Stereographic Projection
given a fnite set $\mathcal{P}$ of points on a $n$-sphere $\mathcal{S}$ and, define a function $f:(s,\mathcal{P})\mapsto\mathbb{R}_0^+$, that maps each point $s$ on $\mathcal{S}$ to the $n$-volume $...
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Volume of intersection of a convex polytope with general affine space
This question generalizes (this question) on the same site.
Let $\Delta^{n}$ denote the $n$-dimensional simplex in $n$ dimensions.
That is, $\Delta^{n}$ is the convex closure of the origin
and the $n$...
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approximate diameter of polytopes in high dimensions
I just came across the following problem:
Let us consider the unit corner of the n-cube
$$
\Delta^n = \left\{(t_1,\cdots,t_n)\in\mathbb{R}^n\mid\sum_{i = 1}^{n}{t_i} \leq 1 \mbox{ and } t_i \ge 0 \...
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Does the Border (Boundary) Points of a convex body make a concave function?
Let $\mathbb{S}$ be a closed and bounded convex body in 2-D with some non-empty intersection with positive quadrant and let it also contain origin. Let $c>0$ be the right-most point on the x-axis ...
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Higher dimensional convex hull
Let $CH(S)$ be a convex hull of a finite set $S$ and denote the set of all the vertices of $CH(S)$ as $Vert(S)$. For a vertex $v \in Vert(S)$, it has an associated set $E(v)$ which is defined as $E(v)=...
user33954
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When is a complete fan a normal fan?
Is there a characterization for when a complete fan in $\mathbf{R}^n$ is the normal fan of a polytope? Thanks!
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Can we uniquely define a graph to have the topology of a polytope via proper edge length selection?
I'll ask you to consider a situation wherein one has a series of edges for a graph, $(e_1, e_2, ..., e_N) \in E$, each with a specifiable length $(l_1, l_2, ..., l_N) \in L$, and the goal is to insure ...
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Intersection numbers on blow ups of toric varieties
Suppose we have a smooth, complete toric varietiy $X_{\Sigma}$ of dimension $n$. Let $\sigma_k \in \Sigma(k)$ a smooth $k$-dimensional cone in $\Sigma$ and suppose we make the toric blow up at the ...
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Vertices of a polytope as algebra generators
I am wondering if the following kind of objects has some name, or are there any studied examples. I apologize for perhaps too specific definition, this is an adoptation of a situation that arises in ...
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Question regard checking convexity by "restriction to any line that intersects the function domain"
Hello all,
I have a question (probably stupid one) about the fact that " A function is convex if and only if it is convex when restricted to any line that intersects its domain".
In Stephen Boyd and ...
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Does a triangulation without fixed simplex property always exist?
Suppose we are given a triangulable topological space $X$. If $X$ has the fixed point property (FPP), then obviously for every triangulation $K$ of $X$ and every simplicial map $f:K\to K$ a simplex $\...
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Can the GUE be thought of as a uniform point in a high-dimensional polytope
I have thought about this question for a long time and could only find partial answers.
The Gaussian Unitary Ensemble (or GUE) is the eigenvalues of a random Hermitian matrix with complex Gaussian ...
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Does there always exist a self dual polytope that contains a given polytope contained in its dual?
Suppose a polytope $P$ is contained in its dual polytope $\tilde{P}$. Does there always exist a polytope $Q$ that contains $P$ and is self dual $Q=\tilde{Q}$? Is there any bound on the minimal number ...
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Polar interpretation of convexity
Let $C$ be a convex polygon in the plane containing the origin, and let $r(\theta)$ for $\theta\in[0,2\pi)$ be a parametrization of its boundary. Is there a condition on $r$ that is equivalent to (or ...
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Terminology and technique for repeated pairwise removal of elements of posets: "Collapsibility" of a "face poset"
Let $P$ be a poset, or partially ordered set. Let $\le$ denote the reflexive order on $P$, and $<$ the corresponding irreflexive order. Let the phrase "a maximal pair" in $P$ refer to an
ordered ...
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Looking for the correct version of a wrong statement from Barvinok's book on convex polyhedra
The book I'm concerned with is "Integer Points in Polyhedra" by A. Barvinok, which, I must say, is turning out to be highly fascinating.
A real finite-dimensional vector space $V$ defines the ...
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What's the most efficient way to solve this euclidean projection on non-negative affine space constraint?
I've come across this convex optimization problem in my research where I need to project a matrix $X_0$ onto a non-negative affine space constraint and box constraints. Concretely,
$X \in \mathbb{R}^{...
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Simplex with edges of length at least s having smallest circumradius
Is it true that of all $n$-simplices with edge lengths greater than or equal to some parameter $s$, the regular simplex with edge lengths $s$ has the smallest circumradius? It seems obvious, but I ...
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Zoll Flat Finsler tori and convex bodies on a starry night
The starry night. The "celestial sphere" is given by set of non-zero vectors in $\mathbb{R}^n$ modulo positive dilations (i.e., $v \equiv w$ if $v = \lambda w$ for some $ \lambda > 0$) and the "...
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Finding a point at which only certain linear functionals are integral
Let $C$ be a full-dimensional rational polyhedral cone in $\Bbb R^d$ with facets $G_1,\ldots,G_n$ . For each $i$, let $h_i$ be an integer-valued linear functional on $\Bbb R^d$ whose kernel is the ...
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Properties of Coefficients of Order Polynomials [closed]
I am working on a problem involving determining the order polynomial $\Omega_P(k)$ of a partial order $P$, which counts the number of order-preserving transformations/maps from $P$ to the $k$-chain (...
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determining a convex set by mixed volumes
For a convex set $K \subset \mathbb{R}^2$ let $\phi_K:$ convexsets in $\mathbb{R}^2 \rightarrow [0,\infty), A \mapsto MV(A,K)$. Where by $MV(A,K)$ I mean the mixed volume of $A$ and $K$ in the ...
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Estimating the Volume of the Metric Polytope
A metric on $n$ points $N$ can be represented as a vector $x \in \mathbb{R}_+^{n \choose 2}$.
For each pair of distinct $i, j \in N$, we have $d(i,j) = d(j,i) = x_{i,j}$. The set of all metrics is ...
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An affine invariant of convex bodies
The following invariants of "pointed" convex bodies (i.e., pairs consisting of a convex body and a distinguished point in its interior) roughly measure how many of its linear images fit between the ...
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Maximizing ratio volume/diameter^n by an affinity
Suppose we have a convex compact body $D\subset \mathbb R^n$. We can try to apply affine transformation keeping the volume and decreasing the diameter of $D$.
It is clear that there is a constant $\...
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Efficient algorithm for finding normals of a high dimensional convex hull with few facets
I am looking for the most efficient algorithm to use to, given a set of points in $d$ dimensional space, find the normals of the convex hull of these points, given that I know that the number of ...
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Projecting on a convex compact polytope with special form
Let $E$ be a large sparse $l$-by-$n$ matrix ($l$ and $n$ can be in the billions...) with coefficients in $\{-1, 0, 1\}$: the first row of $E$ is the vector $(1,0,0,\ldots,0) \in \mathbb R^n$, and ...
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Log convexity for the norm of a vector-valued function
Log convexity of various functions defined on the space of Hermitian matrices plays an important role in matrix analysis and probability theory.
Given $v \in \mathbb{C}^n$, $D$ a diagonal matrix with ...
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How many vertices of a polytope can be chopped off to produce a k-vertex facet?
Let P be a simple n-facet d-polytope with facet F, and let F have k vertices. Let H be a halfspace and Q be a simple (n-1)-facet polytope such that H ∩ Q = P.
In terms of k, what is an upper ...
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Integrally closed polytopes from 01-matrices
Let $A$ be a matrix with entries either 0 or 1, where each column contains at least one 1, to remove trivial degenerations.
Let $P$ be the convex hull of all integer vectors $x$ that satisfy $Ax \leq ...
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Does a polytope have a self-indexing shelling?
If $X$ is a smooth projective toric variety and $P \subset \mathbf{R}^n$ is its moment polytope, then a generic linear function on $\mathbf{R}^n$ induces (1) a Morse function on $X$, and (2) a ...
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Inscribed polytopal approximation to a convex body
This question is on the continuation of the post
Approximation of convex body by polytopes
The central problem I am interested is an explicit construction of inscribed polytope with at most $n$ ...
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Distinguishing (possibly lower dimensional) $1$-skeleton of a regular graph inscribed in a sphere
Consider you have two (possibly same) convex $1$-skeleton of a regular graph $A$ and $B$ in $m$-dimensions inscribed in a sphere with possibly exponential number of vertices in $n$-dimension with ...
user76479
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Volume of intersection of a convex polytope with an affine space.
Given an $n$-dimensional polytope $P\subset \mathbb{R}^n$.
For a subset of indices $i_1,\ldots,i_k$ of $\{1,\ldots,n\}$ and reals $a_1,\ldots,a_k$, we denote by $P(x_{i_1}=a_{1},\ldots,x_{i_k}=a_{k})...
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Is every 1-skeleton of a 4-tope Steinitzian?
Call a graph "polyhedral" if it is simple, planar, and 3-connected. For example, the 1-dimensional skeleton of every 3-dimensional convex polytope, regarded as a graph, is polyhedral. By a theoreom ...
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What is the functional form of the projections of a subdimensional polytope?
Let $\mathbf{S}$ be a $m\times n$ matrix, with $m < n$. We define a subdimensional polytope as the space of $n$-dimensional vectors $\mathbf{x}$ that satisfy the following equation:
$$\mathbf{S}\...
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Sampling from a Convex Body with Many Extremal Points
Let $p_{1}, \ldots, p_{N}$ be a collection of points in $\mathbb{R}^{n}$. I would like to sample uniformly from the convex hull of these $N$ points in an `efficienct' way. In my setting, I have $n$ ...
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How many facets can $\{\|D^T x\|_1\leq 1\}$ have?
$\newcommand{\RR}{\mathbb{R}}$Consider $x\in\RR^n$ and $D\in \RR^{n\times p}$ with $p\geq n$ and full rank. My question is:
How many facets can the polytope $ \{x\in\RR^n\ :\ \|D^T x\|_1\leq 1\}$ ...
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A Combinatorial Abstraction for The "Polynomial Hirsch Conjecture"
Consider $t$ disjoint families of subsets of {1,2,…,n}, ${\cal F}_1,{\cal F_2},\dots {\cal F_t}$ .
Suppose that
(*)
For every $i \lt j \lt k$
and every $R \in {\cal F}_i$, and $T \in {\cal F}_k$,
...
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Polytopes whose intersections have few vertices
This is related to one that I asked earlier:
The intersection of two $l_1$ balls
My question: are there any general classes of polytopes $P_1,P_2\subset\mathbb{R}^n$ such that $P_1$ has few ...
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Right-angled polytopes
%This question is motivated by the little discussion here at the bottom.
The following thing are known about hyperbolic right-angled polytopes:
Compact hyperbolic right-angled polytopes do not exist ...
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What does this number tell me about a convex lattice polygon?
EDIT: I realized I'd tricked myself by working with a too special case of $f$, the question is now updated (boundary lattice points replaced vertices).
Suppose I have a convex lattice polygon $P$, ...
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