Questions tagged [lattice-polytopes]

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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
6 votes
0 answers
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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
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4 votes
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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
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3 votes
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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 ...
Daniel Sebald's user avatar
3 votes
1 answer
670 views

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
1 vote
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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 vote
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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
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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
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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
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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
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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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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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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
1 vote
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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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