Questions tagged [lattice-polytopes]
The lattice-polytopes tag has no usage guidance.
32
questions
3
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1
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670
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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
173
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 ...
0
votes
1
answer
114
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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
77
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 ...
1
vote
0
answers
55
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
127
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. ...
10
votes
3
answers
310
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 ...
1
vote
0
answers
101
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
87
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
252
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
185
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
103
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
239
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
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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
561
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?
...
3
votes
1
answer
282
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
...
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
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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-...
6
votes
1
answer
308
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'...
1
vote
1
answer
115
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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 ...
12
votes
1
answer
407
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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 ...
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,...
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 ...
1
vote
1
answer
103
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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 ...
3
votes
1
answer
206
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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_{\...
3
votes
2
answers
307
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 ...
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
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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
737
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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 ...