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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questions with no upvoted or accepted answers
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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 $T \in {\cal F}_k$,
...
24
votes
0answers
2k views
Why do polytopes pop up in Lagrange inversion?
I'd be interested in hearing people's viewpoints on this. Looking for an intuitive perspective. See Wikipedia for descriptions of polytopes and the Lagrange inversion theorem/formula (LIF) for ...
19
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486 views
“Japanese Theorem” on cyclic polygons: Higher-dimensional generalizations?
A beautiful theorem known as the Japanese Theorem (Wikipedia, MathWorld)
says that, no matter how one triangulates a cyclic (inscribed in a circle) polygon,
the sum of the radii of the incircles is ...
15
votes
0answers
242 views
realization spaces of 3-dimensional polytopes
It is a well-know result (Steinitz, 1922) that the realization space of 3-dimensional convex polytopes with fixed combinatorics is contractible.
A proof of this theorem can be found for instance in ...
14
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0answers
191 views
Have there been further developments on this scheme for polytope approximations to the unit ball of $\ell_p^n$?
A long time ago I happened to look at, and save (on a floppy disk!) for future reading, a copy of the following article:
W. T. Gowers, Polytope approximations of the unit ball of $l^n_p$.
In Convex ...
14
votes
1answer
523 views
Inequalities for marginals of distribution on hyperplane
Let $H = \{ (a,b,c) \in \mathbb{Z}_{\geq 0}^3 : a+b+c=n \}$. If we have a probability distribution on $H$, we can take its marginals onto the $a$, $b$ and $c$ variables and obtain three probability ...
14
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0answers
242 views
Is combinatorial automorphism of symmetric convex polytope always antipodal?
The question is formulated in the title. More precisely, if $P$ is an origin-symmetric convex polytope in $\mathbb{R}^d$, and $f$ is a bijective transform of the set of the vertices of $P$, which ...
14
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0answers
240 views
Polytopes with few vertices and few facets
I recently realized that, for fixed $\alpha$ and $\beta$, the number of (combinatorial types of) $d$-polytopes with $\leq d+1+\alpha$ vertices and $\leq d+1+\beta$ facets is bounded by a constant that ...
14
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0answers
468 views
Does every convex polyhedron have a combinatorially isomorphic counterpart whose angles between edges are rational multiples of $\pi$?
After reading these very interesting questions, I came up with another one:
Does every convex polyhedron have a combinatorially isomorphic counterpart whose angles between all pairs of edges meeting ...
13
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240 views
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 ...
12
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Reciprocity (Ehrhart-style) for real polytopes?
Is there some sense in which the well-known Ehrhart reciprocity law for rational, convex, polytopes can be extended to any convex polytope with arbitrary real vertices?
In other words, given any ...
12
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0answers
1k views
Constructive aspects of Caratheodory's theorem in convex analysis
Let me paraphrase Caratheodory's theorem in a probabilistic setup:
Let $X$ be a real-valued random variable. For $k = 1, \ldots, m$, let $f_k: \mathbb{R} \to \mathbb{R}$ be a continuous function such ...
11
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0answers
307 views
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 ...
11
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0answers
353 views
Lower Bound on the Volume of Certain Polytopes
Given a partition $\rho\in\mathcal{P}(n)$ with $k$ blocks
$$
\rho=\{B_1,B_2,\ldots,B_{k}\}
$$
we can define the set of equations
$$
E_{i}:\sum_{j \in B_{i}}{x_{j-1}}=\sum_{j \in B_{i}}{x_j}\quad\text{...
10
votes
0answers
294 views
Bi-spherical polyhedra
Bicentric polygons have been studied: a polygon all of whose vertices lie on its
circumcirle, and whose incircle is tangent to every edge:
I have not been able to find a comparable literature ...
9
votes
0answers
145 views
How to correctly state Cauchy's rigidity theorem?
Cauchy's rigidity theorem is usually cites briefly as
Any two (convex, 3-dimensional) polyhedra with pairwise congruent faces are themselves congruent.
As a more formal generalization to general ...
9
votes
0answers
183 views
Positivity of coefficients of Ehrhart polynomial of n-Tetrahedron
A set of positive integers $d_1, \dots, d_n$ describe two n-dimensional closed lattice tetrahedron:
$$ T = \left\{ (x_1, \dots, x_n) \in \mathbb{R}^n: \sum_{i=1}^n \frac{x_i}{d_i} \leq 1 \textrm{ and ...
9
votes
0answers
158 views
Is every planar point set the projection of vertices of a neighborly 4-polytope?
More exactly, written in coordinates, I'm curious to know if for every point set $(x_i,y_i)$ there are $(x_i,y_i,z_i,w_i)$ that are vertices of a neighborly polytope.
This problem comes from a simple ...
9
votes
0answers
262 views
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 ...
9
votes
0answers
722 views
Counting Lattice Points in Real Polytopes
Suppose one did have an exact formula for the number of $\mathbb{Z}^n$-lattice points intersecting an arbitrary dilate of a (not necessarily rational) finite, closed and convex $n$-polytope. As a ...
8
votes
0answers
191 views
intriguing Polytope
Define $E_{i,j} \in \mathbb{R}^{n \times n}$ to be the canonical basis (that is all elements set to zero except the entry $i,j$ )
let the bloc matrix $M \in \mathbb{R}^{n^2 \times n^2}$ defined by :
...
8
votes
0answers
155 views
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 ...
8
votes
0answers
133 views
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 $P$, ...
8
votes
0answers
135 views
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\}$ ...
7
votes
0answers
75 views
Can solutions to Thomson's problem have pentagons?
Thomson's problem asks for the minimum energy configuration for $N$ electrons on a sphere (refs: https://en.wikipedia.org/wiki/Thomson_problem, https://sites.google.com/site/adilmmughal/...
7
votes
0answers
116 views
Probability of landing inside the convex hull of previously sampled points
Let $\{X_i\}_{0\leq i\leq\infty}$ be i.i.d. random vectors in $\mathbb{R^d}$.
I would like to show that the probability of one point being in the convex hull of the others goes to one with the number ...
7
votes
0answers
234 views
Integral representations of finite groups and lattice point geometry
This contains both a reference request, and a specific problem.
Let $K$ be a finite group, and let $\theta: K \to {\rm GL}(d,{\bf Z})$ be a (faithful) group representation over the integers. Consider ...
7
votes
0answers
850 views
Maximum volume cross-section of a hypercube
This is surely well known, but:
Q1. What is the $(d{-}1)$-dimensional polytope
that realizes the maximum volume cross-section of a unit hypercube
by a $(d{-}1)$-dimensional hyperplane?
...
7
votes
0answers
487 views
Geometry of the metric cone
Let us say that two metrics $d$ and $d_0$ on a set $X$ are related if there exist positive constants $0 < \alpha \leq \beta$ such that
$$
\alpha \,\left(d_0(x,y) + d_0(y,z) - d_0(x,z)\right) \leq
...
7
votes
0answers
4k views
Convex hulls of compact sets
Let $A$ be a compact set in a separable Hilbert space $H$, and let $\bar A$ denote its convex hull. Is $\bar A$ compact?
7
votes
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331 views
A question about a blue fan and a red fan and their common refinement
Is the following conjecture true?
Conjecture: Let $M_1$ be a red map and let $M_2$ be a blue map drawn in general position on $S^n$, and let $M$ be their common refinement. There is a vertex $w$ of $...
7
votes
0answers
593 views
When should a moment polytope have “smooth” faces?
A codimension $d$ face of a polytope is called rationally smooth if it lies on only $d$ facets, because this is exactly the condition for the corresponding toric variety to have only orbifold ...
6
votes
0answers
134 views
An inequality related to the numbers of faces of polytopes with d+2 facets
I would like to prove an inequality related to the number of $k$-faces of two $d$-polytopes with $d+2$ facets; see (1) below.
Let $r>0$, $s>0$, $t\ge 0$, and $d\ge 2$ be such that $d=r+s+t$. We ...
6
votes
0answers
202 views
Complexity of scissors congruence?
Where is the complexity of the problem 'Given two bounded compact convex integral polyhedra in $\mathbb R^n$ presented by $O(poly(n))$ integer valued halfspaces given by linear inequalities with ...
6
votes
0answers
88 views
Tiling with Horn's polytopes
Let $n\ge2$ be an integer. Consider the hyperplane $H_n$ of ${\mathbb R}^n$ defined by the equation $x_1+\cdots+x_n=0$ and then the sector $P_n\subset H_n$ defined by the inequalities $x_1\le\cdots\le ...
6
votes
0answers
145 views
Cut locus on a hypercube
Inspired by the question, "Shortest path connecting two opposite points on a cube":
Q. What does the cut locus with respect to one corner of a hypercube
in $\mathbb{R}^d$ look like?
"The cut ...
6
votes
0answers
98 views
Rational $d$-simplices
Define a rational $d$-simplex as a simplex in $\mathbb{R}^d$
such that the measure of all its $k$-dimensional faces, $k \ge 1$, is rational.
So a rational triangle has rational edge lengths and ...
6
votes
0answers
105 views
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 ...
6
votes
0answers
171 views
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 ...
6
votes
0answers
214 views
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$ ...
6
votes
0answers
448 views
Minimum solid angle and aspect ratio of an $n$-simplex
In computational geometry and other fields, it is of interest to have degeneracy measures for shapes of simplices, which quantitatively seperate the regular simplex from degenerate simplices.
In two ...
6
votes
1answer
195 views
Triangulations of convex surfaces
Let $M$ be a smooth closed positively curved surface in Euclidean 3-space, $T$ be a geodesic triangulation of $M$, and $E$ be the edge graph of the convex hull of vertices of $T$.
It is easy to see ...
5
votes
0answers
59 views
Which polytopes can be deformed while keeping their edge-lengths?
Let $P\subset\Bbb R^d$ be a convex polytope (a convex hull of finitely many points). Lets call it flexible, if it can be continuously deformed while
keeping its combinatorial type, and
keeping its ...
5
votes
0answers
109 views
How does a map from permutahedra to associahedra factor through multiplihedra?
Let $P_i$ denote permutahedra, $K_i$ associahedra and $J_i$ multiplihedra. In their famous paper on operadic diagonals, Saneblidze and Umble use a projection $p_i: P_i \to K_{i+1}$ which factors as $...
5
votes
0answers
99 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 ...
5
votes
0answers
164 views
Is there a well-established terminology for polyhedra/polytopes?
I got confused lately. It seems like in the metric context a polyhedron tends to mean an intersection of a finite number of half-spaces, while a polytope is a convex hull of a finite set of points. At ...
5
votes
0answers
101 views
Lattice paths in polytopes
Let $P$ be a polytope in $\mathbb{R}^n$. Let $A_ix = b_i$ be the defining equations of its codimension $1$ faces. Is there an algorithm or some kind of criterion to decide if the lattice points inside ...
5
votes
0answers
126 views
Face structures of chain polytopes
For a finite poset $P$ the chain polytope $\mathscr C(P)\subset\mathbb{R}^P$ consists of such $g$ that $g(p)\ge 0$ for all $p\in P$ and $$g(p_1)+\ldots+g(p_n)\le 1$$ for any chain $p_1<\ldots<...
5
votes
0answers
291 views
Algorithm to express a point from a H-polyhedron as convex combination of extreme points
Let $P\subset\mathbb{R}^n$ be a convex polyhedron described as an intersection of hyperspaces, that is,
$$P:=\{\boldsymbol{x}: A\boldsymbol{x} \leq \boldsymbol{b}\}$$
Let $\boldsymbol{x} \in P$. We ...
5
votes
0answers
226 views
Biggest (or large) rectangle in a polytope
I need an efficient method to construct a (hyper)rectangle inside a polytope with a lot of dimensions (say $100 < d < 1000$). Ideally I'd want the biggest possible rectangle, but as I don't ...
