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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56
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2answers
6k views

The amplituhedron minus the physics

Is it possible to appreciate the geometric/polytopal properties of the amplituhedron without delving into the physics that gave rise to it? All the descriptions I've so far encountered assume ...
34
votes
14answers
8k views

Open problems in Euclidean geometry?

Which are some (research level) open problems in Euclidean geometry ? (Edit: I ask just out of curiosity, to understand how -and if- nowadays this is not a "dead" field yet) I should clarify a bit ...
23
votes
10answers
11k views

Algorithm for finding the volume of a convex polytope

It's easy to find the area of a convex polygon by division into triangles, but what is the optimal way of finding the volume of higher-dimensional convex bodies? I tried a few methods for dividing ...
7
votes
7answers
3k views

Uniformly Sampling from Convex Polytopes

How to choose a point uniformly from a convex polytope $P \subset [0,1]^n$ defined by some inequalities, Ax < b ? (Here A is an m-by-n matrix, x is n-by-1 and b is m-by-1.) I imagine that you ...
12
votes
2answers
1k views

Combinatorics of the Stasheff polytopes

First a little background for those unaware. The Stasheff polytopes (or associahedra) are certain convex polytopes that arise in the theory of $A_\infty$-algebras. There is one polytope for each $n\...
18
votes
5answers
906 views

Is there a midsphere theorem for 4-polytopes?

The (remarkable) midsphere theorem says that each combinatorial type of convex polyhedron may be realized by one all of whose edges are tangent to a sphere (and the realization is unique if the center ...
21
votes
3answers
2k views

How many different numbers can be obtained as product of first $n$ natural numbers?

Let m and n be natural numbers, and consider the set of all possible products of m (not necessarily distinct) elements from the set $\{1,2,\ldots,n\}$, that is consider the set $\{1^{a_1} \cdot 2^{...
17
votes
1answer
496 views

Egalitarian measures

A question I got asked I while ago: If $T$ is a triangle in $\mathbb R^2$, is there a function $f:T\to\mathbb R$ such that the integral of $f$ over each straight segment connecting two points in the ...
6
votes
3answers
2k views

Random Sampling a linearly constrained region in n-dimensions…

Hi, So here is my problem: Given a nonlinear, discontinous, cost function $f(x_1,x_2,..,x_N)$ along with linear constraints $x_n \ge 0, \forall n$ $x_n \le c_n$ and $\sum_{n=1}^N x_n = 1$ find an ...
8
votes
1answer
207 views

Tighter Caratheodory on the moment curve?

The moment curve is the set of points of the form $$(t,t^2,t^3,...,t^n) \in R^n$$ Let $M$ be the portion of the moment curve where $t\in [0,1]$, and let $\overline{M}$ be the convex hull of $M$. ...
16
votes
1answer
328 views

Higher dimensional generalization of: Any quadrilateral tiles the plane?

Any (non-self-intersecting) quadrilateral tiles the plane.     (MathWorld image.) Q. What is the strongest known generalization of this statement to higher dimensions? I.e., $\mathbb{...
11
votes
2answers
856 views

Efficiently determine if convex hull contains the unit ball

Given a set of $n$ points in $\mathbb{R}^d$, is there an algorithm to determine if the convex hull contains the unit ball centered at the origin in polynomial time? The convex hull itself might have ...
10
votes
2answers
799 views

An integrality question about expressing an integer as a product of numbers below $n$

Let $n\ge 2$ be a natural number. Suppose that $N$ is a natural number, composed only of primes below $n$, and that can be expressed as $$ N= \prod_{j=1}^{n} j^{x_j} $$ where $x_1$, $\ldots$, $x_n$...
2
votes
0answers
72 views

Relating face polytopes of permutohedra to integer partitions

The OEIS entries A019538, A049019, and A133314, relate a refinement of the face polynomials of the permutohedra (A049019) to partition polynomials (A133314) defined by multiplicative inversion of an ...
15
votes
1answer
374 views

Does every convex polyhedron have a combinatorially isomorphic counterpart whose all faces have rational areas?

Does every convex polyhedron have a combinatorially isomorphic counterpart whose faces all have rational areas? Does every convex polyhedron have a combinatorially isomorphic counterpart whose edges ...
22
votes
13answers
4k views

Generalizations of the Birkhoff-von Neumann Theorem

The famous Birkhoff-von Neumann theorem asserts that every doubly stochastic matrix can be written as a convex combination of permutation matrices. The question is to point out different ...
20
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 ...
10
votes
1answer
2k views

Characterizing faces of 3-dimensional polyhedra. (Related to Victor Eberhard's Theorem [1890]:)

Originally asked by Ali Dino Jumani; EXTENSIVELY EDITED by David Speyer. The previous version was a bit confused, but Steven Sivek and Graham, in the comments, figured out what was going on. G. C. ...
24
votes
2answers
553 views

Complete the following sequence: point, triangle, octahedron, . . . in a dg-category

Let $\mathcal C$ be a pre-triangulated dg-category (or a stable $\infty$-category, if you wish). An object $X$ in $\mathcal C$ gives a "point": $$X$$ A morphism $X\xrightarrow f Y$ in $\mathcal C$ ...
6
votes
0answers
149 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 ...
17
votes
2answers
636 views

“Derived” polyhedra and polytopes

The notion of derived polygon is natural and leads to remarkable convergence. Start with a polygon, and replace it by locating a point on every edge a fraction $\alpha$ between the two endpoints. For ...
17
votes
2answers
799 views

About a Delzant polytope. (In particular dodecahedron)

Hi. I have a question. Definition. Delzant polytope $P$ is a rational convex simple polytope with the smooth condition. Here, "smooth" means that for each vertex $v$, the $n$ edges containing $v$ ...
13
votes
0answers
220 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 ...
9
votes
2answers
546 views

Guises of the Stasheff polytopes, associahedra for the Coxeter $A_n$ root system?

Richard Stanley keeps a famous running compilation of different guises of the celebrated Catalan numbers. The number of vertices of the associahedron is one instantiation among the multitude, and the ...
14
votes
1answer
615 views

Totally rational polytopes

Define a convex polytope in $\mathbb{R}^d$ as totally rational (my terminology) if its vertex coordinates are rational, its edge lengths are rational, its two-dimensional face areas are rational, etc.,...
8
votes
3answers
841 views

On maximal regular polyhedra inscribed in a regular polyhedron

Let T, C, O, D, or I be regular tetrahedron, cube, octahedron, dodecahedron, and icosahedron, respectively. Suppose that the outer polyhedron have edge-length 1. For example, it's easy to prove that ...
7
votes
3answers
1k views

Area of cross-section (at midpoint perpendicular to longest diagonal) in the unit cube of dimension N

Take a unit cube (of side 1) in N dimensions. Construct the cross-section at the midpoint of the longest diagonal. What is the area of this N-1 dimensional region? I can compute this, but it would be ...
5
votes
1answer
845 views

Linear transformation of a polyhedron

Is there a simple proof that shows: Linear transformation of a $\mathcal{H}$-polyhedron (i.e. the intersection of finitely many closed half-spaces) is a $\mathcal{H}$-polyhedron. Minkowski sum of ...
5
votes
1answer
213 views

Approximation of convex body by polytopes

In a recent survey paper, Approximation of convex sets by polytopes, Bronstein claims that under Hausdorff distance $\rho_H$, for every convex body $U$, $$\rho_H(U,\mathcal{P}_n)\leq c(U) n^{-2/(d-1)},...
2
votes
2answers
316 views

Bound on the (anticanonical) degree of toric Fano varieties

Does there exists a universal constant $C \geq 1$ such that if $X$ is any a smooth, toric, Fano $n$-dimensional manifold admitting a Kähler-Einstein metric, then its anticanonical degree $(-K_X)^n$ ...
14
votes
1answer
308 views

Smallest regular simplex containing the unit cube in $R^n$

What is the length $e_n$ of the edge of the smallest $n$-dimensional regular simplex $S_n$ containing the $n$-dimensional unit cube $Q_n$? In particular, is there $n$ such that $e_n<\sqrt{2}(n+...
14
votes
1answer
490 views

3D models of the unfoldings of the hypercube?

There are (apparently) 261 distinct unfoldings of the 4D hypercube, a.k.a., the tesseract, into 3D.1 These unfoldings (or "nets") are analogous to the 11 unfoldings of the 3D cube into the plane.2 ...
13
votes
2answers
3k views

Weighted area of a Voronoi cell

Let $X = \{ x_1,\dots,x_n\} $ denote a set of $n$ points in the unit square $S = [0,1]\times[0,1]$, and let $w = \{w_1,\dots,w_n\}$ denote a set of weights corresponding to the $n$ points in $X$. ...
6
votes
2answers
205 views

Which unfoldings of the hypercube tile 3-space: How to check for isometric space-fillers?

Recently Mark McClure constructed and displayed the 261 unfoldings of the hypercube (tesseract) in response to the question, "3D models of the unfoldings of the hypercube?": The first 9 unfoldings ...
6
votes
2answers
180 views

Gaussian and the convex hull of moment curves

Let $c_1,\dots, c_d$ be the first $d$ moments of the standard normal distribution. Does the point $(c_1,\dots, c_d)$ lie in the convex hull of the set $\{(t,t^2,\dots,t^d)\colon t\in[-b,b]\}$, for a ...
6
votes
2answers
656 views

“MultiCatalan numbers”

Could anyone provide a reference for the following (sort of) generalization of Catalan numbers: the multinomial coefficient $$ \binom{2k_1+3k_2+4k_3+...}{k_1+2k_2+3k_3+...,k_1,k_2,k_3,...} $$ is ...
4
votes
0answers
1k 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?
2
votes
2answers
546 views

Extreme points of transportation polytope

I'm interested in $n \times m$ joint probability tables with prescribed row and column marginals. Such tables form a convex set known as the transportation polytope. What are the extreme points of ...
2
votes
0answers
96 views

The intersection of two $l_1$ balls

Let $B_1$ and $B_2$ be two balls in $\mathbb{R}^n$ with respect to the $l_1$ norm that have different radii and different centers. Is there an upper bound for the number of vertices that $B_1\cap B_2$...
1
vote
2answers
384 views

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 $m$ ...
8
votes
3answers
438 views

Positivity of Ehrhart polynomial coefficients

Are there any results stating that a given family of convex polytopes have Ehrhart polynomials with non-negative coefficients? What methods are available for proving such a property for some family ...
7
votes
1answer
166 views

Convex deltahedra in higher dimensions

There are eight convex polyhedra whose faces are equilateral triangles, so-called deltahedra:        (Image from here) Q. Have the equivalent higher-dimensional ...
6
votes
2answers
995 views

Finding the convex combination of vertices which yields an inner point of a polytope

Given a convex polytope $P\in \mathbb{R}^n$, and a point $x\in P$, Caratheodory's theorem gives us that there exists a set of at most $n+1$ vertices of $P$, such that $x$ is a convex combination of ...
4
votes
2answers
261 views

point in polytope

Suppose I have the convex hull $P$ of a finite collection of points in $\mathbb{R}^d,$ and I want to see whether a point $p$ is contained in $P.$ This is a standard (some would say the standard linear ...
4
votes
5answers
494 views

Lattice points in dilated polytopes and sumsets

Let $P$ be an integral polytope, that is, the convex hull of some points in $\mathbb{N}^d$. Let $p_1,\dots,p_m$ be all lattice points in $P$. Question: What is the condition on $P$ that guarantees ...
4
votes
1answer
196 views

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 "...
3
votes
2answers
128 views

Polytope with indegree-increasing property.

I have a question about a simple polytope. I am worried that my question would be inappropriate for mathoverflow. So I am sorry that I am ignorant of combinatorics. Let $\mathcal{P}$ be a simple ...
2
votes
2answers
268 views

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})...