**17**

votes

**5**answers

452 views

### How many unit simplices are needed to cover a unit $n$-cube?

The volume of an $n$-dimensional simplex of unit edge length is
$$V(n) = \frac{\sqrt{n+1}}{n! 2^{n/2}} \;,$$
so at least $\lceil 1/V(n) \rceil$ such simplices are needed to cover the unit $n$-cube.
...

**3**

votes

**0**answers

101 views

### 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 ...

**3**

votes

**1**answer

128 views

### convex decompositions of the sphere

Consider a decomposition of the sphere $S^n$ into convex pieces (that is, every cell of the cell decomposition is convex, in particular, is contained in a hemisphere). Consider the $k$-skeleta of ...

**3**

votes

**0**answers

83 views

### 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 ...

**2**

votes

**0**answers

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

**0**answers

39 views

### Volume under the intersection of scaled simplices

This is rather specific but I need to compute the volume under the intersection of rescaled simplexes, that is, the volume of the space:
$\left\{x \in \mathbb{R}^n|\sum_i c_{ki} |x_i| \leq1\; k = 1 \...

**3**

votes

**2**answers

91 views

### algorithm of polytope

Basically, we have an incremental sets of vertices
$${V_1} \subset {V_2} \subset ...$$
for each of them, we could build a polytope $${P_i} = Conv({V_i})$$
Consequently, we can compute
$${F_i} = ...

**1**

vote

**0**answers

74 views

### A version of isotone projection cones

We write $a \succeq b$, where both $a, b \in \mathbb{R}^n$, as a shorthand for $a_i \ge b_i$ for all $1 \le i \le n$. Let $C$ be a closed convex cone in the first orthant of $\mathbb{R}^n$ and denote ...

**2**

votes

**0**answers

64 views

### A convex analysis theorem improvement

John's theorem states that to any full-dimensional symmetric convex set $K\subseteq R^n$ and any Ellipsoid $E\subseteq R^n$ that is centered at origin, there exists an invertible linear map $T$ so ...

**3**

votes

**0**answers

87 views

### 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 ...

**1**

vote

**1**answer

360 views

### Computation of extreme rays of rational polyhedral cones - Hemmecke's project and lift algorithm

I am working on an implementation of Raymond Hemmecke's algorithm for finding generating sets of cones: http://arxiv.org/abs/math/0203105
Unfortunately I am struggling to make the algorithm work on ...

**1**

vote

**1**answer

64 views

### number of affine pieces of linear interpolation of convex functions in high dimension

Consider a convex function $f$ defined on a $d$-dimensional hypercube $[0,1]^d$. Now for fixed $m \in \mathbb{N}$, consider the grids $\mathcal{G}_m=\{(i_1/m,\cdots,i_d/m)\}$ where $i_\alpha\in\{0,1,\...

**3**

votes

**0**answers

78 views

### Which (polytopal) fans/polytopes are secondary?

Let $P$ be a (d-1)-dimensional polytope with $n$ vertices that sits in an affine hyperplane in $\mathbb{R}^d$.
The secondary fan of $P$ is a polytopal fan in $\mathbb{R}^n$ with $d$-dimensional ...

**3**

votes

**2**answers

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 ...

**6**

votes

**2**answers

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 ...

**4**

votes

**2**answers

225 views

### regular polyhedra (and polytopes) in hyperbolic geometry, and generalisations

While there exist regular tesselations of the hyperbolic plane with arbitrary regular polygons, there are no new regular polyhedra in hyperbolic (3D) space. This being quite trivial, it is probably ...

**5**

votes

**1**answer

221 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)},...

**4**

votes

**0**answers

79 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 ...

**14**

votes

**1**answer

495 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
...

**5**

votes

**1**answer

150 views

### Maximal volume of a simplex inscribed in a spherical cap

Let $B_n$ be the $n$-dimensional unit ball, and $B_n(\varepsilon)$ be the spherical cap with height $\varepsilon$ I am interested in the quantity
$$\Gamma:=\sup_{\Delta:\textrm{ inscribed simplex in }...

**1**

vote

**0**answers

55 views

### Conditions on probability measure that generates non-void random polytope

Let $C$ be a non-void compact convex set in $\mathbb{R}^d$, and $\nu$ a probability measure on $C$. Then under what conditions on $C$ and $\nu$, the following statement is true: If $X_1,\cdots,X_{d+1}$...

**6**

votes

**1**answer

114 views

### two-dimensional sections of polyhedral cones

Given a polyhedral cone, its intersection with any two-dimensional plane is either a polygon or a region enclosed by a polygonal curve. Is it a characterization of polyhedral cones? Does there exists ...

**5**

votes

**0**answers

74 views

### Is every planar point set be projections 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 $(x_i,y_i,z_i,w_i)$ be vertices of a neighborly polytope.
This problem comes from a simple ...

**6**

votes

**0**answers

151 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 ...

**1**

vote

**1**answer

104 views

### 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}^{...

**2**

votes

**1**answer

169 views

### Is mean width a Dehn invariant?

Let $P \subset \mathbf{R}^3$ be a convex polyhedron. Let $rP$ be the dilation of $P$ by the positive real number $r$.
The Dehn invariant of $P$ is an element of the weird real vector space $\mathbf{...

**2**

votes

**1**answer

266 views

### Sampling point uniformly at random satisfying equality constraints

First of all, I apologize in advance if the question has already been asked in some way on this site and/or if there is a widely known solution to this problem.
The description of my problem is ...

**0**

votes

**0**answers

31 views

### Looking for a homogeneous function with some properties

I'm looking for a 1-homogeneous function $\pi \colon \mathbb{R}^n_{\geq 0} \to \mathbb{R}$ satisfying the following properties:
1) $\pi$ is not concave. This is equivalent to the fact that there ...

**5**

votes

**1**answer

355 views

### Examples of Polyhedra with Large Shadows

Let $P \subseteq \mathbb{R}^n$ be a polyhedron described by $\mathcal{O}(n^{c_1})$ inequalities, where $c_1$ is a constant. Moreover, let $M\colon P \to \mathbb{R}^2$ be a linear mapping. I'm looking ...

**12**

votes

**4**answers

655 views

### What is the number of equitriangulations of the n-cube?

I wonder if this question has been considered before and if anything is known. My search attempts have failed so far.
Let's consider the n-dimesnional cube, $[0,1]^n$, and let's call a simplex with ...

**5**

votes

**1**answer

206 views

### Non-closed geodesics on a convex polyhedron in $\mathbb{R}^3$

Let $P$ be the surface of a closed convex polyhedron in $\mathbb{R}^3$.
Q. Does every non-closed geodesic $\gamma$ fill $P$ densely?
Of course $\gamma$ cannot pass through a vertex of $P$, but ...

**7**

votes

**1**answer

188 views

### A polytope with a bound on the sum of any $k$ variables

Let $2\le k\le n-1$ and define the polytope
$$P_k(n) = \lbrace (x_1,\ldots,x_n) \in \mathbb{R}^n :
-1\le x_{i_1}+\cdots +x_{i_k} \le 1 \text{ for all } 1\le i_1\lt\cdots\lt i_k\le n\rbrace.$$
There ...

**4**

votes

**2**answers

252 views

### Marked chain polytope, has this been studied?

Fix $n$ and consider the polytope given by the inequalities
$$x_i\leq x_j, \text{ and } 0 \leq x_i \leq a_i \text{ for all } 1\leq i<j \leq n,$$
where $a_i \leq a_i\leq \dots \leq a_n$ are fixed ...

**3**

votes

**1**answer

216 views

### Dehn-Sommerville relations for $\Delta$-complexes

Let $M$ be a closed, triangulated manifold of dimension $m$ and $K(M)$ be its triangulation. Let $f_i$ denote the number of $i$-simplices of $K(M)$. As proved by Klee the face numbers satisfy the ...

**1**

vote

**1**answer

150 views

### Linear map of Zonotopes [closed]

Consider a linear system with a map $A: \mathbb{R}^n \rightarrow
\mathbb{R}^m$ such that $y = A x$ with $n \geq m$
The input space $x$ is constrained by a zonotope set $\mathcal{X}
\subseteq \mathbb{...

**10**

votes

**1**answer

326 views

### What's the volume of $\{x\in[0,1]^n|\sum x_i\le t\}$ for real $t$?

EDIT: Thanks for your answers and comments. There is indeed a classical easy formula, given by Pietro Majer (with a simple nice proof) in his answer below.
Given $x\in\mathbb{R}^n$, $x_i$ denotes ...

**44**

votes

**2**answers

2k views

### The view from inside of a mirrored tetrahedron

Suppose you were standing inside a regular tetrahedron $T$ whose
internal face surfaces were perfect mirrors.
Let's assume $T$'s height is $3{\times}$ yours, so that your
eye is roughly at the ...

**12**

votes

**0**answers

186 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 ...

**6**

votes

**1**answer

331 views

### Regular unimodular triangulation for a certain simplex

Consider an $n$-simplex with vertices given by
$(0,0,\dots,r_i,r_{i+1},\dots,r_n)$ where $r_1,\dots,r_n$ are given natural numbers,
and $i=0,1,\dots,n+1$.
Does this simplex admit a regular, ...

**8**

votes

**0**answers

103 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

**3**answers

441 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 ...

**1**

vote

**0**answers

115 views

### Looking for the manuscript “Uniform polytopes” by N. Johnson

The manuscript Uniform Polytopes (1991) by Norman Johnson is cited in the wikipedia page on uniform polytopes (http://en.wikipedia.org/wiki/Uniform_polytope).
Is there an electronic copy of this ...

**17**

votes

**4**answers

612 views

### The limit of edge-midpoint convex polyhedra

Starting with a convex polyhedron $P_1 \subset \mathbb{R}^3$,
replace that with $P_2$, the convex hull of the midpoints of the edges of $P_1$.
Continuing this process, we obtain a ...

**9**

votes

**2**answers

550 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 ...

**4**

votes

**1**answer

112 views

### Are all marked order polytopes normal?

Richard Stanley showed that order polytopes have a unimoudlar triangulation.
In particular, this implies that they are integrally closed/normal.
One can generalize order polytopes to marked order ...

**4**

votes

**2**answers

351 views

### Convex Sets and Nearest Neighbors

For a set $S \subseteq \mathbb{R}^n$ and a point $x \in \mathbb{R}^n$, let $c_S(x)$ be the point $s \in S$ that minimizes $\|x-s\|$ if such a point exists and is uniquely determined. It is known that $...

**0**

votes

**2**answers

117 views

### Information needed to distinguish combinatorially isomorphic polytopes (up to affine equivalence)

I originally posted this question on Stack Exchange, thinking it perhaps does not qualify as "research-level" but it received no answers... hopefully someone here can help.
The title pretty much ...

**6**

votes

**1**answer

208 views

### Some polytopes in $\mathbb R^n$ whose vertices have coordinates 1, -1 or 0

Let $n$ and $k$ be positive integers with $k\leq n$.
Let $P(n,k)$ be the convex hull in $\mathbb R^n$ of the $2^k {n \choose k}$ vectors whose exactly $k$ coordinates belong to $\{\pm 1\}$ all the ...

**7**

votes

**2**answers

212 views

### Pictures of the von Neumann polytope

Are there any graphic portrayals of von Neumann polytopes in low dimensions?

**16**

votes

**1**answer

332 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{...