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Questions tagged [lattice-polytopes]

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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 $\log(n) = \sum_{p |n}v_p(n) \log(p)$. ...
mathoverflowUser's user avatar
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 ...
Sam Hopkins's user avatar
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 ...
Anthony Quas's user avatar
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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,\...
Dac0's user avatar
  • 295
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. ...
giulio bullsaver's user avatar
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. ...
Hugo Chapdelaine's user avatar
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 ...
Hugo Pfoertner's user avatar
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 $X_3=\Conv((-1,-1),(1,0),(0,1))$...
King 's user avatar
  • 21
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 ...
M. Winter's user avatar
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1 vote
0 answers
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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 ...
fagd's user avatar
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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}: |{\...
fagd's user avatar
  • 163
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 ...
Yoav Len's user avatar
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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 ...
Luis Ferroni's user avatar
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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 ...
Daniel Sebald's user avatar
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 ...
Daniel Sebald's user avatar
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? ...
Nandakumar R's user avatar
  • 5,611
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 ...
domotorp's user avatar
  • 18.3k
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 ...
Hans's user avatar
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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-...
Joseph O'Rourke's user avatar
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 ...
Joseph O'Rourke's user avatar
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 ...
user6232872's user avatar
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 ...
Sam Hopkins's user avatar
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 ...
Ben C's user avatar
  • 3,363
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 ...
M. Winter's user avatar
  • 12.7k
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'...
M. Winter's user avatar
  • 12.7k
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_{\...
Marc Besson's user avatar
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,...
Per Alexandersson's user avatar
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 ...
Joseph O'Rourke's user avatar
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 ...
B. Gimazid's user avatar
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 $...
Sam Hopkins's user avatar
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 ...
Harry Altman's user avatar
  • 2,575
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 ...
Wlodek Kuperberg's user avatar