Discrete Isoperimetric Inequalities Research Articles

On a Characterization of Lattice Cubes via Discrete Isoperimetric Inequalities

We obtain a characterization of lattice cubes as the only sets that reach equality in several discrete isoperimetric-type inequalities associated with the $L_{\infty}$ norm, including well-known results by Radcliffe and Veomett. We furthermore provide a new isoperimetric inequality for the lattice point enumerator that generalizes previous results, and for which the aforementioned characterization also holds.

Isoperimetric Upper Bound for the First Eigenvalue of Discrete Steklov Problems

We study upper bounds for the first non-zero eigenvalue of the Steklov problem defined on finite graphs with boundary. For finite graphs with boundary included in a Cayley graph associated to a group of polynomial growth, we give an upper bound for the first non-zero Steklov eigenvalue depending on the number of vertices of the graph and of its boundary. As a corollary, if the graph with boundary also satisfies a discrete isoperimetric inequality, we show that the first non-zero Steklov eigenvalue tends to zero as the number of vertices of the graph tends to infinity. This extends recent results of Han and Hua, who obtained a similar result in the case of mathbb {Z}^n. We obtain the result using metric properties of Cayley graphs associated to groups of polynomial growth.

Wulff shape emergence in graphene

Graphene samples are identified as minimizers of configurational energies featuring both two- and three-body atomic-interaction terms. This variational viewpoint allows for a detailed description of ground-state geometries as connected subsets of a regular hexagonal lattice. We investigate here how these geometries evolve as the number [Formula: see text] of carbon atoms in the graphene sample increases. By means of an equivalent characterization of minimality via a discrete isoperimetric inequality, we prove that ground states converge to the ideal hexagonal Wulff shape as [Formula: see text]. Precisely, ground states deviate from such hexagonal Wulff shape by at most [Formula: see text] atoms, where both the constant [Formula: see text] and the rate [Formula: see text] are sharp.

Crystalline and Isoperimetric Square Configurations

AbstractWe present crystallization results for planar atomic interactions governed by two‐ and three‐body terms with the resulting periodicity being that of the square lattice. The emergence of a (square) Wulff shape for ground states is established by showing the optimality of ground‐state configurations in terms of a discrete isoperimetric inequality. Furthermore, an n3/4 law for the deviation from the asymptotic Wulff shape is established with an explicit constant for the leading term. (© 2014 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

A Discrete Isoperimetric Inequality on Lattices

We establish an isoperimetric inequality with constraint by $$n$$n-dimensional lattices. We prove that, among all sets which consist of lattice translations of a given rectangular parallelepiped, a cube is the best shape to minimize the ratio involving its perimeter and volume as long as the cube is realizable by the lattice. For its proof a solvability of finite difference Poisson---Neumann problems is verified. Our approach to the isoperimetric inequality is based on the technique used in a proof of the Aleksandrov---Bakelman---Pucci maximum principle, which was originally proposed by Cabre (Butll Soc Catalana Mat 15:7---27, 2000) to prove the classical isoperimetric inequality.

Which Boolean Functions Maximize Mutual Information on Noisy Inputs?

We pose a simply stated conjecture regarding the maximum mutual information a Boolean function can reveal about noisy inputs. Specifically, let X n be independent identically distributed Bernoulli(1/2), and let Yn be the result of passing X n through a memoryless binary symmetric channel with crossover probability α. For any Boolean function b : {0, 1} n → {0, 1}, we conjecture that I(b(X n ); Y n ) ≤ 1 - H(α). While the conjecture remains open, we provide substantial evidence supporting its validity. Connections are also made to discrete isoperimetric inequalities.

Finite crystallization in the square lattice

This paper addresses two-dimensional crystallization in the square lattice. A suitable configurational potential featuring both two- and three-body short-ranged particle interactions is considered. We prove that every ground state is a connected subset of the square lattice. Moreover, we discuss the global geometry of ground states and their optimality in terms of discrete isoperimetric inequalities on the square graph. Eventually, we study the aspect ratio of ground states and quantitatively prove the emergence of a square macroscopic Wulff shape as the number of particles grows.

A generalization of the Discrete Isoperimetric Inequality for Piecewise Smooth Curves of Constant Geodesic Curvature

The discrete isoperimetric problem is to determine the maximal area polygon with at most ]]> k$ vertices and of a given perimeter. It is a classical fact that the unique optimal polygon on the Euclidean plane is the regular one. The same statement for the hyperbolic plane was proved by K\'aroly Bezdek [1] and on the sphere by L\'aszl\'o Fejes T\'oth [3]. In the present paper we extend the discrete isoperimetric inequality for ``polygons'' on the three planes of constant curvature bounded by arcs of a given constant geodesic curvature.

Discrete isoperimetric inequalities

Discrete isoperimetric inequalities

Discrete Isoperimetric Inequalities and the Probability of a Decoding Error

We derive improved isoperimetric inequalities for discrete product measures on the n-dimensional cube. As a consequence, a general theorem on the threshold behaviour of monotone properties is obtained. This is then applied to coding theory when we study the probability of error after decoding.

Review of Probabilistic Combinatorics and Its Applications

Probablistic arguments have been applied in many areas of combinatorics and theoretical computer science ever since Erdös first used one to prove bounds on Ramsey numbers. Applications range from constructing graphs with properties useful in building communication networks to almost uniform generation of random structures for the purpose of approximately solving intractable counting problems. This book consists of a series of introductory survey articles on topics in probablistic combinatorics and its applications. (The articles are derived from lectures given in one of a series of short courses sponsored by the American Mathematical Society.) The topics covered include random and random-like graphs, discrete isoperimetric inequalities, rapidly mixing Markov chains, and finite Fourier methods. The emphasis throughout the book is on techniques, with sufficient examples to show their usefulness.

The three dimensional polyominoes of minimal area

The set of the three dimensional polyominoes of minimal area and of volume $n$ contains a polyomino which is the union of a quasicube $j\times (j+\delta)\times (j+\theta)$, $\delta,\theta\in\{0,1\}$, a quasisquare $l\times (l+\epsilon)$, $\epsilon\in\{0,1\}$, and a bar $k$. This shape is naturally associated to the unique decomposition of $n=j(j+\delta)(j+\theta)+l(l+\epsilon)+k$ as the sum of a maximal quasicube, a maximal quasisquare and a bar. For $n$ a quasicube plus a quasisquare, or a quasicube minus one, the minimal polyominoes are reduced to these shapes. The minimal area is explicitly computed and yields a discrete isoperimetric inequality. These variational problems are the key for finding the path of escape from the metastable state for the three dimensional Ising model at very low temperatures. The results and proofs are illustrated by a lot of pictures.