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There are finitely many isomorphism classes of reflexive polytopes in dimension n. In dimensions 1,2,3 and 4, the numbers of isomorphism classes are 1, 16, 4319 and 473800776 by the work of Kreuzer ...
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Let $S \subset \mathbb{R}^n$ be an $n$-simplex with integer vertices. Suppose that $S$ does not contain any integer point other than its vertices and that $2S$ contains at least one integer point in ...
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This was motivated by this recent question: Expansion identity for the Eulerian polynomials of the second order Question: For each integer $m \geq 0$, is there some $2m$-dimensional lattice polytope $...
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Consider an $n$-dimensional lattice polytope $P \subset \mathbb{R}^n$. Let ${\bf m}_i$ denote its vertices. Let $v \in \mathbb{R}^n$ denote an arbitrary vector and use it to define a functional by ${\...
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Fix integers $k, m \geq 0$. For each $r=0,1,\ldots,k+m+1$, define $$ a_{k,m,r} = \sum_{j=0}^{\min(k,m)} \binom{k}{j}\binom{m}{j}\sum_{\substack{S \subseteq \{-j,-j+1,\ldots,k+m-j\},\\\#S=k+m+1-r}}\...
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I am studying a large collection of lattice polytopes, all of them being simple and empty. The dimension can be any integer. The dilatation by $2$ gives non-empty polytopes. For many of these ...
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So let $L$ be a full dimensional lattice in $\mathbb{R}^{n}$. Then the Voronoi-cell of the lattice are precisely the points in $\mathbb{R}^{n}$ that are at least as close to the origin, as to any ...
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Casagrande's paper "The number of vertices of a Fano polytope" says that for $P$ a simplicial reflexive polytope of dimension $n$ has no more than $3n$ vertices. The polymake database ...
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In dimension 1, the only smooth projective toric Fano variety is $\mathbb{P}^1$. In dimension $2$, there are 5: $\mathbb{P}^1\times \mathbb{P}^1$, and then successive blow-ups of $\mathbb{P}^2$ at up ...
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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 \mid n}v_p(n) \log(p)...
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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 ...
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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 ...
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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,\...
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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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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. ...
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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 ...
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$\DeclareMathOperator\Conv{Conv}$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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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 ...
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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 ...
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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}: |{\...
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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 ...
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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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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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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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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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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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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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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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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
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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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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
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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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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
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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's ...
M. Winter's user avatar
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3 votes
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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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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
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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
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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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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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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
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18 votes
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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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