相关论文: Optimal Vector Balancing for Zonotopes

The vector balancing constant $\mathrm{vb}(K,Q)$ of two symmetric convex bodies $K,Q$ is the minimum $r \geq 0$ so that any number of vectors from $K$ can be balanced into an $r$-scaling of $Q$. A question raised by Schechtman is whether…

Zonotopes are studied from the point of view of central symmetry and how volumes of facets and the angles between them determine a zonotope uniquely. New proofs are given for theorems of Shephard and McMullen characterizing a zonotope by…

The isotropy constant of any $d$-dimensional polytope with $n$ vertices is bounded by $C \sqrt{n/d}$ where $C>0$ is a numerical constant.

We study the Subset Balancing problem: given $x \in \mathbb{Z}^n$ and a coefficient set $C \subseteq \mathbb{Z}$, find a nonzero vector $c \in C^n$ such that $c\cdot x = 0$. The standard meet-in-the-middle algorithm runs in time…

Voronoi conjectured that any parallelotope is affinely equivalent to a Voronoi polytope. A parallelotope is defined by a set of $m$ facet vectors $p_i$ and defines a set of $m$ lattice vectors $t_i$, $1\le i\le m$. We show that Voronoi's…

There are d-dimensional zonotopes with n zones for which a 2-dimensional central section has \Omega(n^{d-1}) vertices. For d=3 this was known, with examples provided by the "Ukrainian easter eggs'' by Eppstein et al. Our result is…

We give a full classification of vertex-transitive zonotopes. We prove that a vertex-transitive zonotope is a $\Gamma$-permutahedron for some finite reflection group $\Gamma\subset\mathrm{O}(\mathbb R^d)$. The same holds true for zonotopes…

Given $A\subseteq\mathbb Z_n$, the constant $C_A(n)$ is defined to be the smallest natural number $k$ such that any sequence of $k$ elements in $\mathbb Z_n$ has an $A$-weighted zero-sum subsequence having consecutive terms. The value of…

We prove that there is a universal constant $C>0$ with the following property. Suppose that $n\in \mathbb{N}$ and that $\mathsf{A}=(a_{ij})\in M_n(\mathbb{R})$ is a symmetric stochastic matrix. Denote the second-largest eigenvalue of…

In this note we give a polynomial time algorithm for solving the closest vector problem in the class of zonotopal lattices. The Voronoi cell of a zonotopal lattice is a zonotope, i.e. a projection of a regular cube. Examples of zonotopal…

We use probabilistic, topological and combinatorial methods to establish the following deviation inequality: For any normed space $X=(\mathbb R^n ,\|\cdot\| )$ there exists an invertible linear map $T:\mathbb R^n \to \mathbb R^n$ with \[…

Let $A,B\subseteq \mathbb Z_n\setminus\{0\}$. A sequence $S=(x_1,\ldots, x_k)$ in $\mathbb Z_n$ is called an $(A,B)$-weighted zero-sum sequence if there exist $a_1,\ldots,a_k\in A$ and $b_1,\ldots,b_k\in B$ such that…

We prove that there is an absolute constant $ C$ such that for every $ n \geq 2 $ and $ N\geq 10^n, $ there exists a polytope $ P_{n,N} \subset \mathbb{R}^n $ with at most $ N $ facets that satisfies…

Let $A\subseteq \mathbb Z_n$ be a subset. A sequence $S=(x_1,\ldots,x_k)$ in $\mathbb Z_n$ is said to be an $A$-weighted zero-sum sequence if there exist $a_1,\ldots,a_k\in A$ such that $a_1x_1+\cdots+a_kx_k=0$. By a square, we shall mean a…

We prove that each bounded polytope can be represented as a polynomial zonotope, which we refer to as the Z-representation of polytopes. Previous representations are the vertex representation (V-representation) and the halfspace…

The approximation constant $\lambda_{k}(\zeta)$ is defined as the supremum of real $\eta$ such that $\Vert \zeta^{j}x\Vert\leq x^{-\eta}$ for $1\leq j\leq k$ has infinitely many integer solutions $x$. Here $\Vert.\Vert$ denotes the distance…

A plane configuration {v_1,...,v_m} of vectors in {\mathbb R}^2 is said to be balanced if for any index i, the set of the det(v_i,v_j) for j\neq i is symmetric around the origin. A plane configuration is said to be uniform if every pair of…

A wealth of geometric and combinatorial properties of a given linear endomorphism $X$ of $\R^N$ is captured in the study of its associated zonotope $Z(X)$, and, by duality, its associated hyperplane arrangement ${\cal H}(X)$. This…

For a finite abelian group $(G,+)$, the constant $C(G)$ is defined to be the smallest natural number $k$ such that any sequence in $G$ having length $k$ will have a subsequence of consecutive terms whose sum is zero. For a subset…

A parallelotope $P$ is a polytope that admits a facet-to-facet tiling of space by translation copies of $P$ along a lattice. The Voronoi cell $P_V(L)$ of a lattice $L$ is an example of a parallelotope. A parallelotope can be uniquely…