13
votes
Convex lattice polygons with equal area and perimeter
Pick's theorem says these two convex lattice polygons have area
$$i+\frac{b}{2}-1 = 4 + 10/2 -1 = 8 \;,$$
and they both have perimeter $8 + 2 \sqrt{2}$.
You can see I've "bumped out" two ...
- 151k
13
votes
Accepted
Curve with no embedding in a toric surface
A generic curve of genus $5$ is not a hypersurface in a toric surface. This argument is going to use conceptual ideas from Haase and Schicho's paper "Lattice polygons and the number $2i+7$", ...
- 156k
9
votes
Accepted
Volume of convex lattice polytopes with one interior lattice point
This is addressed in the 2013 paper (appeared in Advances in 2015) by Averkov, Krumpelmann, Nill. The give a sharp bound for the volume of a lattice simplex with one interior lattice point (Theorem 2....
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7
votes
Minimum weight triangulation of lattice points in a circle
I would like to propose a suggestion for finding some asymptotic bound, I think it should be $$\frac{1}{2}\cdot (2+\sqrt{2})\cdot \pi r^2.$$
Namely, the ratio of this number to the actual weight will ...
- 28.9k
7
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There are at most four mutually visible lattice points—?
(Inspired by a meta thread on answers given in comments, I am recapping the answer given in the comments (1 2 3) as a CW answer.)
The largest number of mutually visible points in $\mathbb{Z}^d$ is $2^...
7
votes
Accepted
Denominators of rational polytopes in terms of hyperplane coefficients
Let $x$ be a vertex of $\mathcal{P}$. By Cramer's rule, there is an $n \times n$ matrix $C$ such that each coordinate of $x$ is an integer multiple of $\frac{1}{|\det(C)|}$, and the absolute value of ...
- 32.1k
6
votes
A rational polytope that is not a 01-polytope?
Rather obvious in retrospect, but all 01-polytopes are inscribed (the vertices lie on a sphere). So if $P$ is non-inscribable, then it can't be a 01-polytope. There are non-inscribable rational ...
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5
votes
Accepted
Convex lattice polygons with equal area and perimeter
This answers question 2 as well. But I think both questions are way more suitable for math.stackexchange.
I add another example because, unlike the first, it has the property that multiplied by $I^{n-...
- 5,103
5
votes
Integer decomposition property with a partial order
A slightly more general combinatorial family that satisfies this is the family of $s$-lecture hall polytopes. These can be thought of as a weighted version of order polytopes. For a reference see ...
- 85.6k
5
votes
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Edges of the contact polytope of the Leech lattice
Using the unimodular scaling of the Leech lattice, the length of each minimal vector is $\sqrt{4}$. Fixing a particular minimal vector $u$, the remaining minimal vectors $v$ are:
1 vector $v$ with $\...
- 12.4k
5
votes
Accepted
Minimum weight triangulation of lattice points in a circle
The examples shown below for $\ $ $r=4,\ 5,\ $ and $\ 6\ $ illustrate the idea described in Dmitri's answer. Some modifications in the interior plus the choices of edges near the boundary of the ...
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5
votes
Accepted
Are Minkowski sums of upward closed "convex" sets in $\mathbb{N}^k$ still "convex"? (WAS: Comparing mana costs in Magic: The Gathering)
It seems to me that all of the standard counter-examples to the polytope question easily adapt to be counter examples to this question: Take any polytopes in $\mathbb{Z}^{k-1}$ with $(A_0+B_0) \cap \...
- 156k
5
votes
Volume of convex lattice polytopes with one interior lattice point
Depictions of the two lattice polyhedra mentioned so far (by Wlodek Kuperberg & js21):
Left: $(0,0,0),(4,0,0),(0,4,0),(0,...
- 151k
4
votes
Volume of convex lattice polytopes with one interior lattice point
Just to extend the examples so far: Given a non-decreasing list of $n$ integers $a_1 \leq a_2 \leq \cdots \leq a_n,$ consider the simplex with vertices at the origin and the points with all ...
- 30.1k
4
votes
I believe that all facets of a Voronoi-cell of a lattice are centerally symmetric. Is my argument correct? Is this true?
This is a community wiki answer based on the comments by alesia and Richard Stanley.
You already gave the hardest part of the argument. To finish it off, note that translating by $\mathbf{l}$ the ...
4
votes
A rational polytope that is not a 01-polytope?
Every simplicial polytope is rational, hence there are infinitely many rational polytopes in any fixed dimension. In contrast, there are finitely many 0/1 polytopes, since they cannot have more than $...
- 1,502
4
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Unimodality of $f$-vectors of $0/1$-polytopes
I have a partial answer, sadly I can't get fast enough f-vector computations to confirm my suspicions.
Ziegler describes here some simple ways to get non-monotone f-vectors. Most of the constructions ...
3
votes
Accepted
Bound on mutually x-ray-visible lattice points?
The largest number is $g(x,d) = (x+2)^d$, by an analogous argument as for visibility: if the differences of all coordinates of two points $A,B$ are divisible by $x+2$, then there are at least $x$ ...
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3
votes
Volume of convex lattice polytopes with one interior lattice point
this is a great question and fairly good answers are known for arbitrary dimensions. (I think there are works for $d=3$ but I am not sure.) The relevant papers are
On lattice polytopes having ...
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3
votes
A source for $01$-polytopes
There's Ziegler's 1999 Lectures on 0/1-Polytopes, a 45 page survey on the arXiv. It is also the lead chapter in Polytopes - Combinatorics and Computation, Birkhäuser, 2000.
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3
votes
Accepted
Convex Hulls of Demazure Modules
If I understand the question correctly, there is a nice description of the faces. They show up as so called reduced Kogan faces.
However, taking the convex hull is not the right operation,
as ...
- 15.8k
3
votes
Accepted
Property of convex polygons on integer lattice structures
Consider the convex polygon with vertices $(0,0),(5,1),(4,4),(0,11),(-4,4),(-5,1)$. There are no lattice points on its boundary other than the six vertices. If you take any two that aren't adjacent, ...
- 1,097
2
votes
Integer decomposition property with a partial order
Have you checked the family of marked order polytopes? These include the classical Gelfand-Tsetlin polytopes, and I think I can construct such a partial order in case of GT-polytopes.
Let $T \in kP_\...
- 15.8k
2
votes
Number of vertices of a lattice reflexive polytope
This is due to a confusion about the notion of "smooth polytopes" in the literature. Casagrande's bound applies to simplicial polytopes. Reflexive polytopes come in pairs where -- if this ...
- 511
2
votes
Convex lattice polygons with equal area and perimeter
My earlier comment "parallelogram and kite" was signaling an infinite family of examples of groups of $m$ convex lattice polygons where all of them from the same group have the same diameter,...
- 4,786
1
vote
Accepted
How can I find the hyperplane passing through a 600-cell
You already could have considered your provided vertices within layers according to their last coord values. Within decreasing order you get:
$(0, 0, 0; 1)$: the single point ...
1
vote
Integer decomposition property with a partial order
Just stumbled upon this question, pardon the necropost. There exist broad families of poset polytopes generalizing order and chain polytopes. These are known to have your IDP$\le$ property.
The first ...
- 3,513
1
vote
Problem with the vertices of a convex quadrilateral on integer lattice
I thought a few minutes on fedja's remark about a parallelogram contained inside the quadrilateral. It took me time to realize that you could "slide" an edge up with one point staying on one of the ...
- 13k
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