All Questions
Tagged with convex-polytopes lattices
8 questions with no upvoted or accepted answers
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Lattice paths in polytopes
Let $P$ be a polytope in $\mathbb{R}^n$. Let $A_ix = b_i$ be the defining equations of its codimension $1$ faces. Is there an algorithm or some kind of criterion to decide if the lattice points inside ...
- 501
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Lattice points inside a (n-dimensional) tetrahedron
Hi, overflowers.
I was interested in a sharp lower bound for the number of lattice points (say, integral lattice points) inside the tetrahedron defined by the coordinate hyperplanes and $x_1/a_1+...+...
- 306
3
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maximizing number of lattice points with bounded diameter
Suppose I have a lattice $L$ that's just $\mathbb{Z}^k$ but scaled in every coordinate by some (potentially different) real numbers. Now I want to construct a finite set of lattice points $S \subset L$...
- 480
3
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Lattice points in regular simplex
Suppose we are given a regular (closed) simplex $S$ in a vector space $V$ of dimension $n$, whose vertices have integer values. Then for a lattice $L$, is there a sufficient criterion, for $S$ to ...
- 131
3
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Versions of Helly's Theorem for Unbounded Parallelpipeds
I'm studying Frobenius splitting of toric varieties based on Sam Payne's characterization of Frobenius splitting ("Frobenius splittings of toric varieties" (2008)) and I came across a sort of Helly's ...
- 31
2
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Can computers take uniform samples from a polytope?
For each $r \in \mathbb N $ write $\mathbb Z/ 10^r = \{a/10^r: a \in \mathbb Z\}$ and $P(r)$ for the lattice $(\mathbb Z/10^r)^N \subset \mathbb R^N$.
Suppose the plane $P \subset \mathbb R^N$ is ...
- 1,955
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Barnes-Wall lattices’ contact polytopes
The contact polytopes of the Barnes-Wall lattices in 1, 2, 4, and 8 dimensions are all uniform polytopes. Is this true in any higher number of dimensions?
- 2,773
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16-cell honeycomb (4D tiling by cross-polytopes)
A 4-dimensional cross-polytope (also called 16-cell) is a regular polytope whose vertices are all permutations of $(\pm1,0,0,0)$. It is known that this body tiles the space $\mathbb{R}^4$ by ...
- 503