Questions tagged [lattice-polytopes]

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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 ...
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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 ...
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1 vote
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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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5 votes
3 answers
488 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? ...
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6 votes
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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 ...
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1 vote
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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 ...
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2 votes
1 answer
92 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-...
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3 votes
1 answer
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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 ...
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1 vote
1 answer
98 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 ...
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9 votes
2 answers
219 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 ...
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11 votes
1 answer
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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 ...
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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 ...
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6 votes
1 answer
238 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'...
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3 votes
1 answer
181 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_{\...
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8 votes
0 answers
155 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,...
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3 votes
2 answers
262 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 ...
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6 votes
1 answer
262 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 ...
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4 votes
0 answers
133 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 $...
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28 votes
1 answer
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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 ...
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17 votes
4 answers
520 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 ...
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