**0**

votes

**0**answers

28 views

### How many triangulations of a polytope contain a given simplex [on hold]

Let $P$ be a full-dimensional convex polytope in $\mathbb R^n$, and let $v_1, \dots, v_m$ be its vertices.
A triangulation of $P$ is a set $\mathcal T$ of simplices, which (i) cover $P$, (ii) are ...

**4**

votes

**0**answers

41 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 ...

**4**

votes

**1**answer

102 views

### Coefficients of Ehrhart polynomials, in the binomial-coefficient basis

Let $P$ be the convex hull of a finite set of points in $\mathbb Z^d$, and $p(n) = \#\{nP \cap \mathbb Z^d\}$ be its Ehrhart polynomial, which is also the Hilbert polynomial of the corresponding ...

**11**

votes

**0**answers

143 views

### Is combinatorial automorphism of symmetric convex polytope always odd?

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 ...

**0**

votes

**0**answers

20 views

### Efficient sampling from a polytope with large number of contraints [duplicate]

As far as I know, the most popular way to sample from a polytope (in H-representation)
\begin{equation}
\mathcal{P} := \{z \in \mathbb{R}^n | (Az)_j \le b_j\; \forall j=1,2,\ldots,m\}
\end{equation}
...

**5**

votes

**1**answer

124 views

### Are the primary parallelotopes classified? (equivalently, Voronoi cells of lattices)

A primary parallelohedron is a polyhedron that can fill space with infinite translated copies.
It is known (e.g., Coxeter, H. S. M. Regular Polytopes, 3rd ed. New York: Dover, pp. 29-30, 1973; or, ...

**1**

vote

**0**answers

40 views

### Number of simplices contained in a convex body

I am interested in the following question:
Given a convex body $K$ in $\mathbb{R}^d$ and an $\epsilon>0$ small enough, how many $d$-simplices $\{D_i\}_{i=1}^m\subset K$ do we need at least so that ...

**3**

votes

**1**answer

140 views

### Polytopes of a set of matrices

What is the convex hull of the set of $n^2\times n^2$ matrices
\begin{matrix}E_{s(1),s(1)} &E_{s(1),s(2)}... & E_{s(1),s(n)}\\
E_{s(2),s(1)} &E_{s(2),s(2)}... & E_{s(2),s(n)}\\
.. ...

**1**

vote

**0**answers

39 views

### 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 ...

**1**

vote

**1**answer

69 views

### Internal edges in Convex Polytopes

Suppose $S\subset{\mathbb R}^n$ is an infinite subset that is in general position which means that the intersection of $S$ with every affine subspace of dimension $d<n$ always contains at most ...

**7**

votes

**2**answers

127 views

### $f$-vector of simple convex polytope via directions of facets

Let $P$ be a simple convex polytope in $\mathbb{R}^d$ (that is, any vertex belongs to exactly $d+1$ facets). Given the collection of outer normals to facets of $P$, combinatorics of $P$ may be ...

**0**

votes

**0**answers

33 views

### log convexity for the norm of 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 ...

**6**

votes

**1**answer

166 views

### What polytope is this? Bounded sums with choice of coefficients

Let $b\gt a\gt 0$ be constants. Define $P_n(a,b)$ to be the set of all $(x_1,\ldots,x_n)\in\mathbb{R}^n$ satisfying
$$ |c_1 x_1 + \cdots + c_n x_n| \le 1$$
for every choice of ...

**0**

votes

**1**answer

37 views

### Mapping n variables to the surface of an n+1-dimensional cross polytope [closed]

I'm looking for a function that maps n variables to points on the surface of an n+1-dimensional cross polytope. For example, given one variable, the function would return a point on the perimeter of ...

**-2**

votes

**2**answers

74 views

### Maximal-Orthogonal Convex Hull (or Maximal-Rectilinear Convex Hull) [closed]

Edit : Consider giving a reason for down vote.
In my research, I have come across a this paper from the Computational Geometry field and I am not able to understand the concept of ...

**2**

votes

**0**answers

58 views

### Testing if a point is inside a convex polytope formed by halfspaces in n-dimension

Assume we have a convex polytope that is formed by the intersection of $k$-halfspaces in $\mathbb{R}^{n}$.
$$
a_{0,0}x^{n-1} + {a}_{0,1}x^{n-2} + ... a_{0,n-1} \leq 0
$$
$$
a_{1,0}x^{n-1} + ...

**5**

votes

**1**answer

161 views

### “Database” of simplicial polytopes/spheres

Reading through various papers on polytopes I have come across really interesting examples of simplical polytopes and non-shellable (or non-PL) simplicial spheres but sometimes it is hard to keep ...

**1**

vote

**0**answers

60 views

### About optimizing a convex function on a hypercube

Given a real valued convex function $g$ on $[-1,1]^n$, let $f$ be the restriction of it on the hypercube $\{-1,1\}^n$. I want to find a vertex on the hypercube $\{-1,1\}^n$ on which either (1) $f$ ...

**3**

votes

**0**answers

34 views

### 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 ...

**3**

votes

**0**answers

35 views

### “Spherelike” embedding for smooth Fano polytope

Let $\mathbb{R}^n$ be a vector space with Euclidean inner product and $\mathbb{Z}^n \subset \mathbb{R}^n$ be a lattice. Let $P$ be a full dimension convex lattice polytope in $\mathbb{R}^n$ containing ...

**5**

votes

**1**answer

68 views

### Volume satisfying inequality constraints (simplex subset)

Is there a way to find the volume of the "feasible region" of a standard simplex satisfying simple range constraints?
$x_1+x_2+...+x_n = 1$
$a_1 \le x_1 \le b_1$
$a_2 \le x_2 \le b_2$
$...$
$a_n \le ...

**1**

vote

**1**answer

68 views

### 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 ...

**9**

votes

**2**answers

108 views

### Generate polyhedra by collapsing vertices of a polyhedron

I am looking for basic information about the following idea:
(I) Consider a square. By collapsing two adjacent vertices, we obtain a triangle.
(II) Consider a three-dimensional cube. By collapsing a ...

**1**

vote

**0**answers

59 views

### Choosing the weights of a Voronoi diagram — is this function always the gradient of another function?

This question is related to the earlier question Weighted area of a Voronoi cell . As in that question, let $X = \{ x_1,\dots,x_n\} $ denote a set of $n$ points in the unit square $S = ...

**1**

vote

**0**answers

82 views

### 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 ...

**3**

votes

**1**answer

63 views

### convex hull of the set of permutations with one cycle

is there a way to describe the convex hull of the set of permutation matrices with exactly one non-trivial cycle?
or maybe I should ask for the convex hull of cycle matrices :
let $(i_{1},..,i_{k})$ ...

**5**

votes

**0**answers

60 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 ...

**3**

votes

**2**answers

96 views

### Has anyone developed a technique to generate a polytope given (possibly redundant) inequality constraints? [closed]

I've found a few papers that deal with removing redundant inequality constraints for linear programs, but I'm just trying to find the vertices for a feasible region, given a set of inequality ...

**1**

vote

**0**answers

51 views

### Finding optimal linear transformation for intersection of convex polytopes

I previously posted this on MathSE and am now trying here.
I have the following situation, as shown in the following diagram:
$W=\{w_i\}_{i=1..|W|}$ is a set of vertices (show in diagram in ...

**6**

votes

**1**answer

153 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 $M$.
...

**0**

votes

**0**answers

35 views

### Under which conditions is the union of conic hulls of sets in a cartesian product equal to $\mathbb{R}^N$?

Question: Under which conditions on $A, B\in\mathbb{R}^{N\times N}$ is the function $f: \mathbb{R}^N\mapsto \mathbb{R}^N$,
$$f(v) = A[v]_+ + B[-v]_+$$
surjective? Here $[.]_+$ is an elementwise ...

**5**

votes

**2**answers

97 views

### Inscribed parallelotope in a $d$-simplex

The problem setup is simple: Given a $d$-simplex $\Delta_d:=\{(x_1,\cdots,x_d):x_i\geq 0,\sum_i x_i\leq 1\}$, can we construct a finite sequence of parallelotope $A_i$ so that $\Delta_d=\cup_{i=1}^N ...

**1**

vote

**0**answers

39 views

### 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 $n$ ...

**3**

votes

**1**answer

183 views

### Find the minimum distance between two convex hulls

We work over $\mathbb{R}^N$. Let $\mathbf{P}_1$ denote the hyperplane constructed using $N$ points, each of which is on a different axis (there are $N$ axes). We denote by $\mathbf{P}_2$ the convex ...

**5**

votes

**0**answers

55 views

### Measure-minimizing simplex with fixed inradius

Let $\Delta^n$ be an $n$-simplex in $\mathbb{R}^n$. Let $V$ be the volume and $r$ the inradius (radius of the inscribed sphere) of $\Delta^n$. There is a well-known result that
$$
V \geq ...

**1**

vote

**0**answers

35 views

### An optimality condition for the corners of convex polytopes?

Let $H,H'\subset\mathbb{R}^m$ be two hyperplanes with unit normal-vector,
and let $P\subset\mathbb{R^m}$ be a convex polytope (defined via its corners $v_0, ... , v_n$, where $n\ge m$).
Let's ...

**0**

votes

**0**answers

46 views

### Finding a movement taking out of a convex set

There is a convex set $S$ as the hull of M points in an D-dimensional Euclidean space and a point $\vec P$ in the set. Then, there is a set of vectors $\vec W$ taking the form $\vec W=\sum_{i=1}^N ...

**5**

votes

**1**answer

97 views

### How to prove the existence of the polytope in $\mathbb{R}^d$ with a given number of faces, minimizing the isoperimetric ratio?

This is the isoperimetric type question. We know that in $\mathbb{R}^d$, balls are the sets that minimize the isoperimetric ratio $\frac{S^{d}}{V^{d-1}}$, where $S$ is the surface area and $V$ is the ...

**4**

votes

**0**answers

89 views

### Prove or disprove a claim about covering a polytope by convex polytopes in a certain way

Here is the claim:
Given a polytope $K$ in a unit ball in $\mathbb{R}^d$, there exists a universal constant $C(d)>0$ depending only on $d$ and a countable collection of convex polytopes ...

**7**

votes

**4**answers

108 views

### Uniform Sampling Subject to Linear Equalities and Non-Negativity Constraint

I'm trying to sample uniformly on the intersections of faces of several simplicies, with all coordinates being non-negative. That is, given constraints
$$A\vec{w}=\vec{b} \ \ and \ \ \vec{w} \geq ...

**9**

votes

**2**answers

285 views

### Book on the tetrahedron

Does anybody know of a book containing "all you want to know about the tetrahedron"? What you want to know should include basic geometry of the tetrahedron, study of orthocentric tetrahedra, the Monge ...

**5**

votes

**1**answer

166 views

### Measurement of “symmetry” of a convex body

I often hear that the regular simplex is "the least" symmetric convex body, and I've heard that there are some measures of symmetry of a body, that the simplex minimizes.
Could you please explain or ...

**5**

votes

**2**answers

131 views

### Differentiability of polytope shadow areas

Let $P$ be an opaque convex polyhedron containing the origin in $\mathbb{R}^3$,
and let $S$ be an origin-centered sphere strictly containing $P$: $S \supset P$.
For a point $x$ on $S$, let $\sigma(x)$ ...

**0**

votes

**1**answer

112 views

### polynomial expression for counting number of integral points of a set

Let $v_i=a_ie_i\in\mathbb R^d$ and $w_i=b_ie_i\in\mathbb R^d$ for $i=1,\dots,d$ where $e_i$'s are unit vetcors and $a_i,b_i$ are positive integers. Let $$S=conv\{0,rv_i+sw_i:i=1,\dots,d\}.$$
Can we ...

**1**

vote

**0**answers

71 views

### 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 ...

**2**

votes

**1**answer

47 views

### 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 ...

**4**

votes

**2**answers

70 views

### Expressing a convex Polytope as a sublevel set of a function

Given an n-dimensional polytope $P$ in $\mathbb R^n$, Given as a convex hull of a finite set of points, $S$ I would like to construct an expliict formula for a function $f\colon \mathbb R^n \to ...

**2**

votes

**0**answers

139 views

### Find the intersection between two convex hulls, in this specific case

We work over $\mathbb{R}^K$. Let $V$ be the set of vectors whose coordinates take values $0$ or $1$, or equivalently the corners of the unit cube $[0,1]^K$.
Let $d:\{0, \ldots, K\} \to \mathbb{R}_+$ ...

**2**

votes

**0**answers

71 views

### Newton polyhedron and product of ideals

Let $I$ be an ideal generated by monomials $\underline{x}^{\underline{a}_1},\ldots ,\underline{x}^{\underline{a}_s}$ and
$J$ be the ideal generated by $\underline{x}^{\underline{b}_1},\ldots ...

**2**

votes

**2**answers

231 views

### Cardinality of non-integer points in the translation of the Minkowski sum of convex hull.

Let $\operatorname{conv}(a_1,\ldots,a_m)$ denote the convex hull of $\{a_1,\ldots,a_m\}$. Let $\mathbb{Z}_+=\mathbb{N}\cup\{0\}$ and $\mathbb{Q}_+$ denotes the positive (inluding 0) rational numbers. ...