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Questions tagged [lattice-polytopes]

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6
votes
1answer
186 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 ...
6
votes
0answers
78 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
2answers
215 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 ...
-1
votes
0answers
27 views

Does Barvinok's algorithm count modulo $q$ in $O(polylog (q))$ word size?

Let $Ax\leq b$ be a polyhedron in $n$-dimensions and $m$ constraints and $q>1$ be an integer. The number of points in the polyhedron could be exponential in $n$ and $m$ while $q\ll nm$ could be ...
4
votes
0answers
107 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 $...
27
votes
1answer
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 ...
16
votes
4answers
346 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 ...