Tagged Questions
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
0
votes
0answers
37 views
additive support functions of a convex set
Let $K \subset \mathbb{R}^d$ be a compact, convex set. It could be uniquely determined by its support function (for $u$ on the unit-sphere $S^{d-1}$), given by
$$h_K(u) = \sup \{ \sum_{i=1}^d x_i ...
15
votes
4answers
359 views
+100
Is the diameter of a centrally symmetric convex body realized by a pair of antipodal points?
Let $S \subset \mathbb{R}^n$ be the boundary of a centrally symmetric convex body and provide $S$ with the geodesic metric given by its embedding in Euclidean space (i.e., the distance between two ...
3
votes
0answers
124 views
Dead Flies Problem [duplicate]
If a set of points in the plane contains one point in each convex region of
area 1, then can it have finite density?
what is the density of the points? In my understanding, it means the average ...
4
votes
1answer
96 views
Generalized permutahedron and random polytopes
The Birkhoff polytope $B_n$ is defined as the convex hull of the set of permutation matrices, which gives us the set of doubly stochastic matrices. A concept which is intimately related is that of the ...
5
votes
0answers
74 views
Geometry of the metric cone
Let us say that two metrics $d$ and $d_0$ on a set $X$ are related if there exist positive constants $0 < \alpha \leq \beta$ such that
$$
\alpha \,\left(d_0(x,y) + d_0(y,z) - d_0(x,z)\right) \leq
...
0
votes
0answers
82 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 ...
2
votes
0answers
33 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
1answer
61 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
0answers
65 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
1answer
62 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:
...
2
votes
1answer
80 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
0answers
105 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
0answers
49 views
vertex representation and half-space representation of a polytope [migrated]
i'm not a mathematician. So, my question may be stupid. Hope that it won't make you angry.
Is there any mathematical relationship between the vertex representation and the half-space representation ...
1
vote
0answers
56 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
1answer
52 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 ...
6
votes
0answers
109 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 ...
3
votes
1answer
309 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
1answer
325 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
1answer
191 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
1answer
71 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
1answer
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, ...
5
votes
0answers
64 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 ...
2
votes
1answer
36 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 ...
2
votes
0answers
132 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
5answers
388 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
1answer
103 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 ...
3
votes
1answer
117 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 ...
2
votes
0answers
67 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
1answer
215 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 ...
6
votes
2answers
368 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 $x_1$, $\ldots$, ...
20
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 ...
6
votes
1answer
95 views
Maximizing ratio volume/diameter^n by an affinity
Suppose we have a convex compact body $D\subset \mathbb R^n$. We can try to apply affine transformation keeping the volume and decreasing the diameter of $D$.
It is clear that there is a constant ...
2
votes
0answers
89 views
Definition of self-dual polytope
Given a d-polytope $P$ we define the c-dual polytope as $P^\ast = \{y\in R^d \mid x\cdot y\geq -c, \forall x\in P\}$. Then I say that a polytope is c-polar self-dual if $P=P^\ast$. I cannot find this ...
5
votes
0answers
148 views
4D polytope analogues of the icosahedron/Rogers-Ramanujan continued fraction relationship?
The formula for the j-function which employs polynomial invariants of the icosahedron,
$$j(\tau)=-\frac{(r^{20} - 228r^{15} + 494r^{10} + 228r^5 + 1)^3}{r^5(r^{10} + 11r^5 - 1)^5}$$
where,
...
1
vote
1answer
102 views
2-neighborhood of a simplex
Let $\Delta$ be an $n-1$-simplex in ${\mathbb R}^{n-1}$. For each vertex $v$ of $\Delta$ let $H_v$ be the hyperplane through $v$ and parallel to the opposite facet. By 2-neighborhood of a simplex I ...
15
votes
2answers
411 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 ...
6
votes
0answers
83 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 ...
9
votes
1answer
193 views
Why do convex polytope options constrict with dimension, rather than expand?
There are an infinite number of regular polygons in the plane,
five regular polyhedra,
six regular polytopes in $\mathbb{R}^4$,
and then three regular polytopes in every dimension $d > 4$.
There ...
8
votes
1answer
120 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 ...
12
votes
0answers
120 views
realization spaces of 3-dimensional polytopes
It is a well-know result (Steinitz, 1922) that the realization space of 3-dimensional convex polytopes with fixed combinatorics is contractible.
A proof of this theorem can be found for instance in ...
4
votes
0answers
147 views
Wasserstein distance, convex polytopes and extreme points
Let us consider convex polytopes with $K$ extreme points in $\mathbb{R}^d$. Let $\mathbf{P}$ be such polytope, and let $\text{ext}(\mathbf{P})$ denote its set of extreme points, so that $\mathbf{P} = ...
0
votes
0answers
38 views
Edge versus vertex assignment in graphs and convex relaxtions
Consider a graph $G = (V,E)$. Let $x \in \{-1,1\}^V$ be a label assignment to vertices of the graph and $z \in \{-1,1\}^E$ be a label assignment to edges of the graph. We say that $z$ is compatible ...
1
vote
0answers
75 views
Distance function from the origin to the boundary of a convex polytope
Let $K\subseteq \mathbb R^n$, $n\ge 3$ be a convex bounded polytope containing the origin in its interior. Consider the unit sphere with geodesic distance $(\mathcal S^{n-1},d)$ and define the ...
2
votes
0answers
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 ...
2
votes
3answers
295 views
Repeating an operation infinitely makes any convex $n$-gon a regular $n$-gon?
For any convex $n$-gon $P_{0,1}P_{0,2}\cdots P_{0,n}$, let us consider the following operation :
Operation : Let $k=0,1,\cdots$. Take $n$ points $P_{k+1,i}\ (i=1,2,\cdots,n)$ outside of $n$-gon ...
4
votes
1answer
170 views
A construction related to scissors congruence
I was thinking about the following some time ago. My question is whether such things have been studied before.
Let $E_n$ be the abelian group with a generator for each (bounded) euclidean polytope of ...
6
votes
1answer
499 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 ...
1
vote
2answers
348 views
Vertices of a Polytope
Given two convex polytopes $P,Q$ such that $P\subset Q$. We are given that all the vertices of $P$ are also vertices of $Q$ and all the facets defining planes of $Q$ are also facets defining planes of ...
48
votes
1answer
4k 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 ...
4
votes
2answers
174 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 ...
