# Questions tagged [lattice-polytopes]

The lattice-polytopes tag has no usage guidance.

32
questions

3
votes

1
answer

677
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### Infinite dimensional lattice for integers and the Riemann hypothesis?

It is known that for each finite set of primes $p$ we have: $\log(p)$ are linear independent over the rational numbers.
We have $\log(ab) = \log(a)+\log(b)$ and $\log(n) = \sum_{p |n}v_p(n) \log(p)$.
...

4
votes

1
answer

174
views

### Denominators of rational polytopes in terms of hyperplane coefficients

Let $\mathcal{P}$ be a convex polytope in $\mathbb{R}^n$ given in the form $\mathcal{P} = \{ x \in \mathbb{R}^n\colon A x\leq b \}$. Suppose that the entries of $A$ and $b$ are integers. Then it is ...

1
vote

0
answers

79
views

### Counting Voronoi cells generated by lattice points

I am working on a problem in dynamical systems where I need to count Voronoi cells arising from nearest neighbours to a subset of the lattice. (See the picture below for an example: the shaded region ...

0
votes

1
answer

114
views

### How can I find the hyperplane passing through a 600-cell

I have a 600-cell, whose coordinates are given by
$$\begin{array}{ccc}
\text{8 vertices} & \left(0,0,0,\pm1\right) & \text{all permutations,}\\
\text{16 vertices} & \frac{1}{2}\left(\pm1,\...

1
vote

0
answers

62
views

### Facets of polytopes and toric morphisms

To every convex lattice polytope $P$ is associated a toric variety $X_P$, which can be realized as a projective variety.
Consider a facet $f$ of $P$, i.e. a codimension one boundary of the polytope.
...

1
vote

0
answers

131
views

### Partial exponential sums over lattice points of lattice cones

Consider the usual lattice $M:=\mathbb{Z}^2\subseteq\mathbb{R}^2$, and
let $v_1,v_2\in\mathbb{Z}^2\subseteq\mathbb{R}^2$ be two non-zero lattice points which are $\mathbb{Z}$-linearly independent. ...

1
vote

0
answers

102
views

### Upper bound on the diameter of a convex lattice n-gon with a given area

Given the area $A$ of a strictly convex polygon with $n$ vertices with integer Cartesian coordinates, there are usually several non-equivalent polygons. The relationship between the area, the number ...

1
vote

0
answers

89
views

### All 3-dimensional symmetric reflexive polytopes

$\DeclareMathOperator\Conv{Conv}$I am finding all 3-dimensional symmetric reflexive polytopes. To do so, first, we know that all 2 dim symmetric reflexive polytopes are $X_3=\Conv((-1,-1),(1,0),(0,1))$...

4
votes

2
answers

253
views

### A rational polytope that is not a 01-polytope?

A 01-polytope is the convex hull of some points $S\subseteq\{0,1\}^n$. I wonder, which polytopes can be represented (combinatorially) as 01-polytopes? There are polytopes that cannot have rational ...

1
vote

0
answers

42
views

### How to construct lattices with largest possible number of Voronoi relevant lattice vectors?

Let M be the generator matrix of a $N$ dimensional lattice, and $V$ the set of Voronoi relevant vectors. The Voronoi cell for the origin can be written as $\text{Vor}_{\bf 0}(M)=\left\{{\bf x}: |{\bf ...

0
votes

1
answer

188
views

### How to find the closest point given the Voronoi relevant vectors?

Let M be the generator matrix of a $N\times N$ lattice, and $\tilde{N}$ the set of Voronoi relevant vectors. The Voronoi cell for the origin can be written as $\text{Vor}_{\bf 0}(M)=\left\{{\bf x}: |{\...

1
vote

0
answers

104
views

### Intersection of lattice polytopes

Is there a way to characterise when the intersection of two or more lattice polytopes is again a lattice polytope? For instance, can you read that property from their Ehrhart polynomials? If it makes ...

3
votes

1
answer

243
views

### Unimodality of $f$-vectors of $0/1$-polytopes

It is known that the face vectors (aka $f$-vectors) of general polytopes need not be unimodal. This even fails for simple or simplicial polytopes, as was shown first by Björner.
My question is if ...

3
votes

0
answers

115
views

### Expanded 24-simplices in the Leech polytope

The vertex coordinate set for the contact polytope of the Leech lattice listed on Wikipedia contains all permutations of:
$\{4,-4,0^{22}\}$
$\{-3,1^{23}\}$
$\{3,-1^{23}\}$
The convex hull of these ...

1
vote

0
answers

48
views

### Lattice deformations of regular polytopes

It is trivial to see that the 24-cell, all hypercubes, and all polytopes with simplicial facets, can be deformed into lattice polytopes, and this blog post implies the same is true for the ...

5
votes

3
answers

564
views

### Convex lattice polygons with equal area and perimeter

A convex polygon all of whose vertices have integer coordinates is a convex lattice polygon.
Do there exist mutually non-congruent convex lattice polygons which have the same area and same perimeter?
...

6
votes

0
answers

98
views

### Zero-area-free embedding of points on the grid

Given $n$, I am looking for the smallest $m$ such that there is an $n$ element subset $S$ of the $m\times m$ grid (i.e., $n$ points with integer coordinates in $[0,m]^2$) such that no matter how one ...

1
vote

0
answers

63
views

### Toric resolution in terms of polytopes

Let $P,Q\subset\mathbb{R}^n$ lattice polytopes such that $P$ and $P'=P+Q$ are smooth polytopes. We obtain the birational morphism $f:X_{P'}\to X_Q$ and I am interested in a criterion when this is a ...

2
votes

1
answer

104
views

### Bound on mutually x-ray-visible lattice points?

Say that two lattice points $a$ and $b$ of $\mathbb{Z}^d$
are $x$-visible to one another if the segment $ab$
contains at most $x$ lattice points (excluding $a$ and $b$).
So $x$-visiblity is "x-...

3
votes

1
answer

285
views

### There are at most four mutually visible lattice points—?

Say that two lattice points $a$ and $b$ of $\mathbb{Z}^2$
are visible to one another if the line segment $ab$
contains no other lattice points.
While exploring lattice polygons all of whose vertices
...

1
vote

1
answer

118
views

### Property of convex polygons on integer lattice structures

Another graduate student and I are working on an research project and are looking for a paper or other source that has a proof for a result about polygons on an integer lattice structure. Suppose you ...

10
votes

3
answers

312
views

### Integer decomposition property with a partial order

Let $\mathcal{P}$ be a convex lattice polytope in $\mathbb{R}^n$. We say that $\mathcal{P}$ has the integer decomposition property (or "is IDP") if for all $k\in \mathbb{N}$ and $\alpha \in ...

12
votes

1
answer

412
views

### Curve with no embedding in a toric surface

I am looking for a smooth proper curve $C$ such that there does not exist any closed embedding $C \to S$ where $S$ is a (normal projective) toric surface.
Since $C$ is smooth I believe it suffices to ...

1
vote

1
answer

103
views

### A source for $01$-polytopes

Can you recommend any books or survey articles on $01$-polytopes, thats is, polytopes with vertices in $\{0,1\}^n$?
I am less interested in random $01$-polytopes, but more in the combinatorial ...

6
votes

1
answer

312
views

### Edges of the contact polytope of the Leech lattice

Let $P\subset\Bbb R^{24}$ be the contact polytope of the Leech lattice, that is, $P$ is the convex hull of the 196,560 shortest vectors of $\Lambda_{24}$.
Question: What are the edges of $P$?
Let'...

3
votes

1
answer

210
views

### Convex Hulls of Demazure Modules

Let $G$ be a semisimple algebraic group over $\mathbb{C}$ and for a highest weight $\lambda$, denote by $V_{\lambda}^w$ the Demazure module associated with $\lambda$ and $w$. More precisely, $V_{\...

8
votes

0
answers

168
views

### An Ehrhart positivity question related to Schur polynomials

Consider the Schur polynomial $s_\lambda(x_1,\dotsc,x_k)$.
It is easy to see from the hook-content formula for counting the number of semi-standard tableaux, that the function
$$
n \to s_{n \lambda}(1,...

3
votes

2
answers

315
views

### Minimum weight triangulation of lattice points in a circle

Let $r$ be a natural number, and consider the $\mathbb{Z}^2$ lattice points
$S$ inside or on the circle $C$ of radius $r$ centered on the origin.
Let $P$ be the convex hull of $S$; so $P$ is inscribed ...

6
votes

1
answer

268
views

### Problem with the vertices of a convex quadrilateral on integer lattice

I made the following observation and I am wondering if it is always true.
Let $x_1$, $x_2$, $x_3$ and $x_4$ be four positive integer points in the plane ($x_i\in\mathbb{Z^2_{\geq 0}}$) forming a ...

4
votes

0
answers

139
views

### Reciprocity for multi-parameter Ehrhart polynomials

In McMullen's 1977 paper "Valuations and Euler-type relations on certain classes of convex polytopes" (https://londmathsoc.onlinelibrary.wiley.com/doi/abs/10.1112/plms/s3-35.1.113), he shows that for $...

28
votes

1
answer

1k
views

### Are Minkowski sums of upward closed "convex" sets in $\mathbb{N}^k$ still "convex"? (WAS: Comparing mana costs in Magic: The Gathering)

This was originally a question about comparing mana costs in Magic: The Gathering, but it's turned into a question about Minkowski sums of upward-closed convex sets in $\mathbb{N}^k$. The original ...

17
votes

4
answers

753
views

### Volume of convex lattice polytopes with one interior lattice point

Let $P$ be a convex polytope in $\mathbb{R}^3$ whose every vertex lies in the $\mathbb{Z}^3$ lattice.
Question: If $P$ contains exactly one lattice point in its interior, what is the maximum possible ...