相关论文: The diameter of lattice zonotopes
A point in the $d$-dimensional integer lattice $\mathbb{Z}^d$ is primitive when its coordinates are relatively prime. Two primitive points are multiples of one another when they are opposite, and for this reason, we consider half of the…
We provide a sharp estimate for the asymptotic number of lattice zonotopes, inscribed in $[0,n ]^d$ when $n$ tends to infinity. Our estimate refines the logarithmic equivalent established by Barany, Bureaux, and Lund when the sum of the…
We show that the largest possible diameter $\delta(d,k)$ of a $d$-dimensional polytope whose vertices have integer coordinates ranging between $0$ and $k$ is at most $kd-\lceil2d/3\rceil$ when $k\geq3$. In addition, we show that…
A lattice (d, k)-polytope is the convex hull of a set of points in dimension d whose coordinates are integers between 0 and k. Let {\delta}(d, k) be the largest diameter over all lattice (d, k)-polytopes. We develop a computational…
In this paper we show that the diameter of a d-dimensional lattice polytope in [0,k]^n is at most (k - 1/2) d. This result implies that the diameter of a d-dimensional half-integral polytope is at most 3/2 d. We also show that for…
A subset of the Hamming cube over $n$-letter alphabet is said to be $d$-maximal if its diameter is $d$, and adding any point increases the diameter. Our main result shows that each $d$-maximal set is either of size at most $(n+o(n))^d$ or…
The diameter of the graph of a $d$-dimensional lattice polytope $P \subseteq [0,k]^{n}$ is known to be at most $dk$ due to work by Kleinschmidt and Onn. However, it is an open question whether the monotone diameter, the shortest guaranteed…
It is shown that the smallest possible distance between two disjoint lattice polytopes contained in the cube $[0,k]^3$ is exactly $$ \frac{1}{\sqrt{2(2k^2-4k+5)(2k^2-2k+1)}} $$ for every integer $k$ at least $4$. The proof relies on…
We construct a hollow lattice polytope (resp. a hollow lattice simplex) of dimension $14$ (resp.$~404$) and of width $15$ (resp.$~408$). They are the first known hollow lattice polytopes of width larger than dimension. We also construct a…
We introduce and study a family of polytopes which can be seen as a generalization of the permutahedron of type $B_d$. We highlight connections with the largest possible diameter of the convex hull of a set of points in dimension $d$ whose…
A theorem of Scott gives an upper bound for the normalized volume of lattice polygons with exactly $i>0$ interior lattice points. We will show that the same bound is true for the normalized volume of lattice polytopes of degree 2 even in…
Let (lambda_d)(p) be the p monomer-dimer entropy on the d-dimensional integer lattice Z^d, where p in [0,1] is the dimer density. We give upper and lower bounds for (lambda_d)(p) in terms of expressions involving (lambda_(d-1))(q). The…
We prove that in each dimension $d$ there is a constant $w^\infty(d)\in \mathbb{N}$ such that for every $n\in \mathbb{N}$ all but finitely many $d$-polytopes with $n$ lattice points have width at most $w^\infty(d)$. We call $w^\infty(d)$…
We construct $d$-dimensional pure simplicial complexes and pseudo-manifolds (without boundary) with $n$ vertices whose combinatorial diameter grows as $c_d n^{d-1}$ for a constant $c_d$ depending only on $d$, which is the maximum possible…
We prove that any d-dimensional zonotope obtained from permutahedron by deleting zone vectors has belt diameter at most 3. Moreover if d is not greater than 6 then its belt diameter is bounded from above by 2. Also we show that these bounds…
We consider how flat a lattice simplex contained in the hypercube $[0,k]^d$ can be. This question is related to the notion of kissing polytopes: two lattice polytopes contained in the hypercube $[0,k]^d$ are kissing when they are disjoint…
We investigate the following question: how close can two disjoint lattice polytopes contained in a fixed hypercube be? This question stems from various contexts where the minimal distance between such polytopes appears in complexity bounds…
The lattice size of a lattice polytope is a geometric invariant which was formally introduced in the context of simplification of the defining equation of an algebraic curve, but appeared implicitly earlier in geometric combinatorics.…
We relate the maximum semidefinite and linear extension complexity of a family of polytopes to the cardinality of this family and the minimum pairwise Hausdorff distance of its members. This result directly implies a known lower bound on…
We determine the limiting density of the largest sum-free subset of the lattice cube $\{1,2,\ldots,n\}^d$ for all $d$, thus resolving the natural conjecture that it is constructed by two appropriate hyperplane slices.