Questions tagged [lattice-polytopes]
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14
questions with no upvoted or accepted answers
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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,...
6
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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 ...
4
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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 $...
3
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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 ...
3
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1
answer
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 $\log(n) = \sum_{p |n}v_p(n) \log(p)$.
...
1
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0
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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 ...
1
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0
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55
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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.
...
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127
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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. ...
1
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101
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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 ...
1
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87
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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 $X_3=\Conv((-1,-1),(1,0),(0,1))$...
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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 ...
1
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103
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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 ...
1
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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 ...
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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 ...