# Tagged Questions

**3**

votes

**1**answer

310 views

### The Stock Market Polytope: Explanation?

Ovidiu Racorean.
"Crossing Stocks and the Positive Grassmannian I: The Geometry behind Stock
Market."
(arXiv Abstract link)
Anyone care to offer a summary of what's going on here?
(The ...

**1**

vote

**1**answer

326 views

### Study of convex polytopes via commutative algebra

Let $P \subset \mathbb{R}^d$ be any convex polytope with integral vertices, and let $M$ be the additive submonoid of $\mathbb{R}^{d+1}$ which is generated by $\{ (v,1) : v \in P \cap \mathbb{Z}^d \}$. ...

**2**

votes

**1**answer

45 views

### Integer point in a non-empty polytope

I have a high-dimensional, non-empty polytope $Ax\geq b$ sitting inside the cube ($0\leq x_i \leq 1$). Is there any general theory or technique to show that this polytope contains an integer point, ...

**2**

votes

**0**answers

77 views

### n-dimensional Delaunay Triangulation of Lattices

I have several questions concerning the Delaunay triangulation of a high dimensional lattice.
Given an $n$-dimensional lattice $L$ and its Delaunay triangulation (partition of $R^n$ into simplices ...

**4**

votes

**1**answer

179 views

### n-simplex in an intersection of n balls

Consider any $n$-simplex, $n \geq 2$. For each edge $(i,j)$, consider $n$-ball $B_{ij}$ such that vertices $x_i$ and $x_j$ are antipodal on this ball. Fix a point $x_0$ in the simplex. The question: ...

**13**

votes

**1**answer

180 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 ...

**8**

votes

**3**answers

492 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 ...

**9**

votes

**2**answers

203 views

### Do maximal polyhedra have algebraic volume?

Is it possible to prove that for every $n > 3$ the maximal possible volume of a convex polyhedron having $n$ vertices inscribed in a sphere of unit radius is an algebraic number?
Update: What ...

**5**

votes

**2**answers

238 views

### Covering convex polygons with inscribed disks

The following problem came up when discussing mapping software (e.g., Google maps) with computer scientists. By $B(c,r)$ I mean the planar disk (open or closed, it doesn't matter) of radius $r$ around ...

**6**

votes

**2**answers

166 views

### Volumes of convex vs non-convex polyhedra with prescribed facets areas

It is known that given a set of Areas $A_f$ and normals $\vec{n}_f$ if $\sum_f A_f \vec{n}_f=0$ exist a unique convex polyhedron with given face areas and normals. (Minkowski theorem - See Alexandrov ...

**9**

votes

**1**answer

290 views

### Needle probing for a convex body

Suppose there is an unknown closed convex body $K$ of
volume vol$(K) = V$ inside the
unit cube $[-\frac{1}{2}, \frac{1}{2}]^d$ in $\mathbb{R}^d$.
You are permitted to probe with a (one-dimensional)
...

**3**

votes

**1**answer

141 views

### 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 ...

**1**

vote

**0**answers

138 views

### Lattice-point enumeration question involving linear combinations of matrices

I would like to know some references to learn more about an answer to this question, if there are any references:
Let $A_1, \dots , A_m$ and $B$ be $n\times n$ symmetric matrices. Let $$S = \{(x_1, ...

**2**

votes

**2**answers

178 views

### Is there a simple test to determine whether a polytope is integral?

It is known that any rational convex polytope expressed as $\{ x\in\mathbb{R}^d : Ax \ge b \}$, where $A\in\mathbb{Z}^{k\times d}$ and $b\in\mathbb{Z}^k$, can be written as the convex hull of finitely ...

**3**

votes

**1**answer

195 views

### Empty convex polytopes for random point sets

I know of the famous results on the Erdős-Szekeres empty convex polygon problem in the plane
(the Happy-Ending Problem), and I know that there are higher-dimensional extensions.
A great source ...

**8**

votes

**1**answer

412 views

### Triangulations of polytopes and tilings of zonotopes

Consider a set $A = \{ a_1,a_2,\ldots, a_n \} $ of vectors in $\mathbb{R}^d$, which lie in a common affine hyperplane. Two convex polytopes may be obtained from $A$, namely the convex hull of the ...

**10**

votes

**0**answers

238 views

### Update to Shephard's “Twenty Problems on Convex Polyhedra”

Forty-three years ago, Geoffrey Shephard published an influential list of open problems
on convex polyhedra.
Progress has been made on several of his problems, and perhaps some have been completely ...

**26**

votes

**2**answers

833 views

### Bodies of constant width?

In two-dimensional case one can generalize figures of constant width as figures which can rotate in a covex polygon.
Here is one example which can be used to drill triangular holes:
I would like to ...

**1**

vote

**0**answers

134 views

### When is the conical hull of a finite set of vectors a subset of the space? (and tilings)

Consider a hypercube in n-dimensions, and take some projection down to an m-dimensional subspace. Now take all vertices and m-1 dimensional facets visible from some direction outside the projection. ...

**10**

votes

**2**answers

1k 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

**0**answers

254 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 ...

**43**

votes

**1**answer

2k views

### two tetrahedra in R^4

It is relatively easy to show (see below) that if we have two equilateral triangles of side 1 in $R^3$,
such that their union has diameter 1, then they must share a vertex.
I wonder whether we have an ...

**4**

votes

**2**answers

225 views

### Finding the Boundary Faces of the Zonohedron

A zonotope is a linear combination of m vectors with coefficients in [0,1]: $Z = \{ \sum \lambda_i v_i : 0 \leq \lambda _i \leq 1 \}$. The fancy way is to say it's the Minkowski sum of line segments ...

**5**

votes

**1**answer

615 views

### Which (semi)regular polyhedra are combinations of two others?

The convex combination of convex polytopes is a convex polytope.
An example in $\mathbb{R}^2$ is that a regular octagon
can be obtained as $\frac{1}{2} S + \frac{1}{2} S'$,
where $S$ is a square and ...

**5**

votes

**2**answers

396 views

### Making 'circles' on a lattice/ Making distinct fractions from partitions of a number

First formulation: discrete geometry
Pick your favourite 2D square lattice (I'm sure we all have one...) and try to place n points 'in a circle' (that is: in general position [no 3 should be ...

**16**

votes

**3**answers

1k views

### Cutting convex sets

Any bounded convex set of the Euclidean plane can be cut into two convex pieces of equal area and circumference.
Can one cut every bounded convex set of the Euclidean plane into an arbitrary number ...

**8**

votes

**6**answers

887 views

### Combinatorial distance ≡ Euclidean distance

Definition: A polytope has property X iff there is a function f:N+ → R+ such that for each pair of vertices vi, vj the following holds:
disteuclidean(vi, vj) = f(distcombinatorial(vi, vj))
with ...