**5**

votes

**2**answers

252 views

### Are shortest halving curves simple closed geodesics?

Let $S$ be a smooth convex surface in $\mathbb{R}^3$
(although my question may as well be asked for the surface of a polyhedron).
Say that $\gamma$ is a shortest halving curve if
(a) it partitions the ...

**1**

vote

**2**answers

224 views

### Volume of normal cone of a simplex (at a vertex)

This question is related to an orthant-type simplex in $\mathbb{R}^n$, which can be defined as
$$ S = \{x\in\mathbb{R}_{+}^n: \sum_{i=1}^nx_i \leq 1\}=\overline{\mbox{conv}}(0,e_1,\ldots,e_n). $$
For ...

**2**

votes

**1**answer

183 views

### intersection of the unit cube and a hyperplane containing the main diagonal

Let $A$ be a linear $m$-dimensional subspace of $\mathbf{R}^n$ $m < n$, containing the point $(1,1,\ldots,1) \in \mathbf{R}^n$,
and consider the intersection of $A$ and the unit cube $\Delta_n$ ...

**3**

votes

**0**answers

54 views

### Minimax-like theorems involving union and intersection of regions in $\mathbb R^d$

From Minimax theorems, we roughly know that if $f(x,y)$ is convex on $X$ and concave $Y$ for both compact $X,Y$, then:
$$
\max_{x\in X}\min_{y\in Y} f(x,y)=\min_{y\in Y}\max_{x\in X} f(x,y).
$$
We can ...

**4**

votes

**1**answer

185 views

### The number of facets of a polyhedron under linear transformation

Consider a (not necessarily bounded) convex polyhedron $P\subset \mathbb{R}^n$ which has $k$ facets.
Let $L:\mathbb{R}^n \to \mathbb{R}^m$ be a linear transformation.
Question1: Is there a fixed ...

**5**

votes

**1**answer

813 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 ...

**3**

votes

**2**answers

187 views

### Unimodular triangulation and Ehrhart polynomials

Let $P$ be a convex lattice polytope. Then it has a polynomial Ehrhart function.
I am interested in what can be said about the Ehrhart polynomial when
$P$ has any of the properties
is integrally ...

**2**

votes

**1**answer

168 views

### Vertices of a polytope as algebra generators

I am wondering if the following kind of objects has some name, or are there any studied examples. I apologize for perhaps too specific definition, this is an adoptation of a situation that arises in ...

**0**

votes

**0**answers

55 views

### Paths on Cartesian products of graphs satisfying linear constraints

Assume integers $d > r > 0$ and a connected graph $G$ with $d$ vertices. Every point on the $r$-fold Cartesian product of $G$ with itself, $G^{\square r}$, is equivalent to a dimension-$d$ ...

**23**

votes

**2**answers

526 views

### Convex hull of total orders

Let $n$ be a positive integer and $\prec$ an arbitray total order on $\{1,\dots,n\}$. I associate to this order a vector $v$ with one coordinate for every pair $(i,j)$ s.t. $1\leq i\neq j \leq n$, by ...

**2**

votes

**1**answer

92 views

### Volume of a polytope with relaxed constraints

Consider a polytope in $n$ dimensions defined by a set of linear constraints:
$$P = \{x \in \mathbb{R}^n : Ax \leq b\}$$
where A is some $m \times n$ constraint matrix, and $b = (b_1,\ldots,b_m)$ is ...

**2**

votes

**0**answers

277 views

### Cavalieri's principle and inversion of the Vandermonde matrix

There are many examples on the Web of the use of Cavalieri's principle in determining areas and volumes of 2-D and 3-D geometrical figures. The Wikipedia link uses the principle as both a proof and ...

**1**

vote

**0**answers

64 views

### Does the set of points minimizing their distance to a multiset of convex polytopes result in a polytope?

Let $\mathbb{R}^n$ be a normed affine space of finite dimension $n$, and $d : \mathbb{R}^n \times \mathbb{R}^n \mapsto \mathbb{R}^+$ be the distance derived from the norm under consideration. A convex ...

**8**

votes

**1**answer

213 views

### Status of Barany's conjecture?

One of Barany's most intriguing conjectures is about the $f$-vectors of convex polytopes. It asks:
Let $P$ be a convex $d$-polytope. Is it always true that $f_k \geq \min(f_0, f_{d-1})$?
A ...

**13**

votes

**0**answers

350 views

### Surface area of convex hull [duplicate]

Let Q be the convex hull of a non-convex polyhedron P. Is it true that the surface area of Q is not greater than the surface area of P?

**2**

votes

**1**answer

144 views

### The Mahler conjecture and non-zonoidal 3-polytopes (4-polytopes)

I have been working on the Mahler conjecture for over a year now and have made some progress for certain classes of convex polytopes and I'm now attempting to write up my results specified to ...

**2**

votes

**1**answer

113 views

### Integrally closed polytopes from 01-matrices

Let $A$ be a matrix with entries either 0 or 1, where each column contains at least one 1, to remove trivial degenerations.
Let $P$ be the convex hull of all integer vectors $x$ that satisfy $Ax \leq ...

**10**

votes

**1**answer

355 views

### doubly-stochastic isomorphisms of graphs

A doubly stochastic matrix that commutes with the adjacency matrix of a graph is a doubly-stochastic automorphism of that graph (definition by Tinhofer 1986). Each (classical) automorphism of a graph ...

**2**

votes

**1**answer

188 views

### Calculate the discrete set of points B which are in the convex hull of the set of points A

This problem is likely best described with the following picture:
Given the discrete set of points $A$ (shown in blue), I wish to calculate the discrete set of points that are contained within the ...

**17**

votes

**0**answers

312 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 ...

**8**

votes

**0**answers

117 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\}$ ...

**5**

votes

**2**answers

228 views

### Triangulations of special polyhedra

Let $A_1,A_2,A_3 \in \mathbb{N}^3$ be three points in space all lying in some plane $x+y+z=d$ where $d$ is a positive integer. If $\{e_1,e_2,e_3\}$ is the standard basis in $\mathbb{R}^3$, we can ...

**2**

votes

**1**answer

201 views

### Looking for a canonical (matroid polytope) subdivision of the hypersimplex

A matroid polytope is the convex hull of the indicator vectors of the bases of a matroid, and a matroid polytope subdivision (MPS) is a polyhedral subdivision of a matroid polytope whose cells are ...

**22**

votes

**5**answers

775 views

### Is the diameter of a centrally symmetric convex body realized by a pair of antipodal points?

Let $S \subset \mathbb{R}^n$ be the boundary of a centrally symmetric convex body and provide $S$ with the geodesic metric given by its embedding in Euclidean space (i.e., the distance between two ...

**3**

votes

**0**answers

142 views

### Dead Flies Problem [duplicate]

If a set of points in the plane contains one point in each convex region of
area 1, then can it have finite density?
what is the density of the points? In my understanding, it means the average ...

**4**

votes

**1**answer

291 views

### Generalized permutahedron and random polytopes

The Birkhoff polytope $B_n$ is defined as the convex hull of the set of permutation matrices, which gives us the set of doubly stochastic matrices. A concept which is intimately related is that of the ...

**5**

votes

**0**answers

143 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
...

**0**

votes

**0**answers

101 views

### Ref request: If an affine subspace $V$ of $\mathbb R^n$ meets an $n$-dimensional polytope $P$, then it meets $P$ in a face of dimension $\le n-\dim V$

Let $P$ and $V$ be, respectively, a bounded full-dimensional polytope and an $m$-dimensional affine subspace of $\mathbb R^n$, and assume $P \cap V$ is non-empty.
Q1. Could you provide me with a ...

**4**

votes

**1**answer

329 views

### Simplex in convex polytope, pulling triangulation

Let $P$ be a convex $d$-dimensional polytope.
I have two questions, related to triangulations of $P$.
Question 1:
Let $p$ be in the interior of $P$. Can I always find a triangulation of $P$,
such ...

**3**

votes

**1**answer

89 views

### Number of lattice polytopes contained in a given lattice polytope?

Given a (convex) lattice polytope, suppose we want to list or count all (convex) lattice polytopes (of the same dimension) contained in it. Are there efficient ways to do this?

**4**

votes

**0**answers

265 views

### Regular cross-sections of a dodecahedron; analogous sections of 4-polytopes

One can intersect a dodecahedron with a plane and
obtain an equilateral triangle, a square, a regular pentagon,
a regular hexagon, and a regular decagon:
...

**1**

vote

**1**answer

87 views

### What is the functional form of the projections of a subdimensional polytope?

Let $\mathbf{S}$ be a $m\times n$ matrix, with $m < n$. We define a subdimensional polytope as the space of $n$-dimensional vectors $\mathbf{x}$ that satisfy the following equation:
...

**3**

votes

**1**answer

147 views

### On a conjecture by Hibi regarding h-vectors

For integral polytopes, it is conjectured (T. Hibi), that if the $h^*$-vector is symmetric, then it is also unimodal (increasing, then non-decreasing).
A non-integral polytope do not, in general, ...

**6**

votes

**0**answers

152 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 $n$ ...

**1**

vote

**0**answers

75 views

### Stronger condition than being a normal polytope?

A polytope $P$ with integer vertices is called normal if
for every $p = \sum_j a_j p_j $ such that $a_j \geq 0$, $\sum_j a_j = k \in \mathbb{N}$,
$p_j$ are vertices of $P$ and $p$ is an integer ...

**2**

votes

**1**answer

70 views

### The minimal number of halfspaces to represent a convex but non strongly convex cone

We say a cone at the origin in $R^n$ means that it is an intersection of finitely many halfspaces, i.e.
$$C=\bigcap_{i\in I}H_i,\text{ where }|I|<\infty.$$
A cone is strongly convex if $C\cap ...

**8**

votes

**0**answers

168 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 ...

**2**

votes

**1**answer

673 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 ...

**2**

votes

**1**answer

379 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 $\{ (v,1) : v \in P \cap \mathbb{Z}^d \}$. ...

**5**

votes

**1**answer

285 views

### Normal polytopes - counterexample?

An integral polytope $P$ is normal if all lattice points inside the integer dilation $kP$ can be expressed as $p_1+p_2+\dots+p_k$, where $p_i \in P$ are lattice points.
I am looking for an example ...

**2**

votes

**1**answer

141 views

### Relation between non-integral polytopes, integrally closed polytopes and polynomial Erhart quasi-polynomials

By lattice points, I will always mean points in $\mathbb{Z}^n$ and all polytopes here are convex rational polytopes.
If $P$ is an integral polytope, the counting function for the number of lattice ...

**2**

votes

**1**answer

72 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, ...

**7**

votes

**1**answer

128 views

### Classifying two-faces of four-polytopes

Motivation: This question is related to my study of hyperbolic Coxeter polytopes. In general, if one put some restrictions on the type of their dihedral angles (say, all dihedral angles are equal to ...

**3**

votes

**1**answer

53 views

### Seeking criteria for “threadable” pairs of centrosymmetric polyhedra

Let $A$ and $B$ be origin-centered centrosymmetric polyhedra in $\mathbb{R}^3$:
"for every point $(x, y, z)$ [...] there is an indistinguishable point $(-x, -y, -z)$."
Say that $A$ and $B$ are ...

**4**

votes

**0**answers

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?

**4**

votes

**5**answers

492 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 ...

**2**

votes

**1**answer

366 views

### Approximation of a convex body by a contained polytope

This question deals with approximating a convex body (a compact convex set of $\mathbb{R}^d$ with non-empty interior) by convex polytopes.
For a given $\delta$, let $n_\delta$ be the number of faces ...

**4**

votes

**1**answer

155 views

### Toric ideal of slice of a polytope?

Given a collection $A:=\{a_1, \ldots ,a_n \}$ of different integer points in $\mathbb{N}^d$, which span an affine hyperplane when viewed in $\mathbb{R}^d$, one can define a toric ideal $I_A$ from a ...

**3**

votes

**0**answers

112 views

### Convex polyhedra jammed in $k$ disjoint holes

For a given convex polyhedron $P \subset \mathbb{R}^3$,
I was imagining finding the optimal "fixing" of $P$ in holes (or jamming them in "mud"),
which led to the following question.
First, scale $P$ ...

**6**

votes

**1**answer

333 views

### Has this generalization of a determinant (assigning multiplicities to the rows) been studied?

I'm working on some questions in tropical geometry, and my problem led me to create the following generalization of a determinant:
Let $A$ be an $m \times n$ matrix with $m \le n$, and positive ...