On the structure of subsets of the discrete cube with small edge boundary

We show that a non-empty subset of an abelian group with a small edge boundary must be large; in particular, if $A$ and $S$ are finite, non-empty subsets of an abelian group such that $S$ is independent, and the edge boundary of $A$ with respect to $S$ does not exceed $(1-γ)|S||A|$ with a real $γ\in(0,1]$, then $|A| \ge 4^{(1-1/d)γ|S|}$, where $d$ is the smallest order of an element of $S$. Here the constant $4$ is best possible. As a corollary, we derive an upper bound for the size of the largest independent subset of the set of popular differences of a finite subset of an abelian group. For groups of exponent $2$ and $3$, our bound translates into a sharp estimate for the additive dimension of the popular difference set. We also prove, as an auxiliary result, the following estimate of possible independent interest: if $A \subset \mathbb Z^n$ is a finite, non-empty downset then, denoting by $w(a)$ the number of non-zero components of the vector $a\in A$, we have \[\frac1{|A|} \sum_{a\in A} w(a) \le \frac12\, \log_2 |A|.\]

We prove an isoperimetric inequality on the discrete cube which is the precise analog of a logarithmic inequality due to Talagrand. As a consequence, the Gaussian isoperimetric inequality is derived.

On a biased edge isoperimetric inequality for the discrete cube

The topic of generating all subsets of a given set occupies an important place in the books on combinatorial algorithms. The considered algorithms, like many other generating algorithms, are of two main types: for generating in lexicographic order or in Gray code order. Both use binary representation of integers (i.e., binary vectors) as characteristic vectors of the subsets.Ordering the vectors of the Boolean cube according to their weights is applied in solving some problems—for example, in computing the algebraic degree of Boolean functions, which is an important cryptographic parameter. Among the numerous orderings of Boolean cubic vectors by their (Hamming) weights, two are most important. The weight ordering, where the second criterion for sorting the vectors of equal weights, is the so-called weight-lexicographic order. It is considered in detail in [2]. Considered here is the second weight ordering, where the vectors of equal weights are arranged so that every two consecutive vectors differ in exactly two coordinates, i.e., they are ordered by minimal change. The properties of this ordering are derived. Based on these, an algorithm was developed that generates the vectors of the Boolean cube in this ordering. It uses a binary representation of integers and only performs additions of integers instead of operations on binary vectors. Its time and space complexities are of a linear type with respect to the number of vectors generated. The algorithm was used in the creation of sequence A351939 in OEIS [8].Keywordsn-dimensional Boolean cubeBinary vectorWeight ordering relationMinimal changeGenerating all subsetsGray codeRevolving door algorithm

We derive fundamental asymptotic results for the expected covering radius <f>$\\rho (X_N)$</f> for <f>$N$</f> points that are randomly and independently distributed with respect to surface measure on a sphere as well as on a class of smooth manifolds. For the unit sphere <f>$\\mathbb {S}^d \\subset \\mathbb {R}^{d+1}$</f>, we obtain the precise asymptotic that <f>$\\mathbb {E}\\rho (X_N)[N/\\log N]^{1/d}$</f> has limit <f>$[(d+1)\\upsilon _{d+1}/\\upsilon _d]^{1/d}$</f> as <f>$N \\to \\infty $</f>, where <f>$\\upsilon _d$</f> is the volume of the <f>$d$</f>-dimensional unit ball. This proves a recent conjecture of Brauchart et al. as well as extends a result previously known only for the circle. Likewise, we obtain precise asymptotics for the expected covering radius of <f>$N$</f> points randomly distributed on a <f>$d$</f>-dimensional ball, a <f>$d$</f>-dimensional cube, as well as on a three-dimensional polyhedron (where the points are independently distributed with respect to volume measure). More generally, we deduce upper and lower bounds for the expected covering radius of <f>$N$</f> points that are randomly and independently distributed on a compact metric measure space, provided the measure satisfies certain regularity assumptions.

We determine all maximum weight downsets in the product of two chains, where the weight function is a strictly increasing function of the rank. Many discrete isoperimetric problems can be reduced to the maximum weight downset problem. Our results generalize Lindsay's edge-isoperimetric theorem in two dimensions in several directions. They also imply and strengthen (in several directions) a result of Ahlswede and Katona concerning graphs with maximal number of adjacent pairs of edges. We find all optimal shifted graphs in the Ahlswede-Katona problem. Furthermore, the results of Ahlswede-Katona are extended to posets with a rank increasing and rank constant weight function. Our results also strengthen a special case of a recent result by Keough and Radcliffe concerning graphs with the fewest matchings. All of these results are achieved by applications of a key lemma that we call the reflect-push method. This method is geometric and combinatorial. Most of the literature on edge-isoperimetric inequalities focuses on finding a solution, and there are no general methods for finding all possible solutions. Our results give a general approach for finding all compressed solutions for the above edge-isoperimetric problems.
 By using the Ahlswede-Cai local-global principle, one can conclude that lexicographic solutions are optimal for many cases of higher dimensional isoperimetric problems. With this and our two dimensional results we can prove Lindsay's edge-isoperimetric inequality in any dimension. Furthermore, our results show that lexicographic solutions are the unique solutions for which compression techniques can be applied in this general setting.

A sequence of finite-dimensional normed spaces is constructed, each with two symmetric bases, such that the sequence of equivalence constants between these bases is unbounded. An essential tool in the proof is the edge-isoperimetric inequality in the discrete cube.

We prove a vertex isoperimetric inequality for the $n$-dimensional Hamming ball $\mathcal{B}_n(R)$ of radius $R$. The isoperimetric inequality is sharp up to a constant factor for sets that are comparable to $\mathcal{B}_n(R)$ in size. A key step in the proof is a local expansion phenomenon in hypercubes.

An isoperimetric inequality for antipodal subsets of the discrete cube

We give some estimates for the minimum projector norm under linear interpolation on a compact set in ${\mathbb R}^n$. Let $\Pi_1({\mathbb R}^n)$ be the space of polynomials in $n$ variables of degree at most $1$, $\Omega$ is a compactum in ${\mathbb R}^n$, $K={\rm conv}(\Omega)$. We will assume that ${\rm vol}(K)>0$. Let the points $x^{(j)}\in \Omega$, $1\leq j\leq n+1,$ be the vertices of an $n$-dimensional nondegenerate simplex. The interpolation projector $P:C(\Omega)\to \Pi_1({\mathbb R}^n)$ with the nodes $x^{(j)}$ is defined by the equations $Pf\left(x^{(j)}\right)=f\left(x^{(j)}\right)$. By $\|P\|_\Omega$ we mean the norm of $P$ as an operator from $C(\Omega)$ to $C(\Omega)$. By $\theta_n(\Omega)$ we denote the minimal norm $\|P\|_\Omega$ of all operators $P$ with nodes belonging to $\Omega$. By ${\rm simp}(E)$ we denote the maximal volume of the simplex with vertices in $E$. We establish the inequalities $\chi_n^{-1}\left(\frac{{\rm vol}(K)}{{\rm simp}(\Omega)}\right)\leq \theta_n(\Omega)\leq n+1.$ Here $\chi_n$ is the standardized Legendre polynomial of degree $n$. The lower estimate is proved using the obtained characterization of Legendre polynomials through the volumes of convex polyhedra. More specifically, we show that for every $\gamma\ge 1$ the volume of the set $\left\{x=(x_1,...,x_n)\in{\mathbb R}^n : \sum |x_j| +\left|1- \sum x_j\right|\le\gamma\right\}$ is equal to ${\chi_n(\gamma)}/{n!}$. In the case when $\Omega$ is an $n$-dimensional cube or an $n$-dimensional ball, the lower estimate gives the possibility to obtain the inequalities of the form $\theta_n(\Omega)\geqslant c\sqrt{n}$. Also we formulate some open questions.

In this paper we consider the influences of variables on Boolean functions in general product spaces. Unlike the case of functions on the discrete cube, where there is a clear definition of influence, in the general case several definitions have been presented in different papers. We propose a family of definitions for the influence that contains all the known definitions, as well as other natural definitions, as special cases. We show that the proofs of the BKKKL theorem and of other results can be adapted to our new definition. The adaptation leads to generalizations of these theorems, which are tight in terms of the definition of influence used in the assertion.

In this paper we first introduce quermassintegrals for free boundary hypersurfaces in the 
\n(
\nn
\n+
\n1
\n)
\n-dimensional Euclidean unit ball. Then we solve some related isoperimetric type problems for convex free boundary hypersurfaces, which lead to new Alexandrov–Fenchel inequalities. In particular, for 
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\n=
\n2
\n we obtain a Minkowski-type inequality and for 
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\n=
\n3
\n we obtain an optimal Willmore-type inequality. To prove these estimates, we employ a specifically designed locally constrained inverse harmonic mean curvature flow with free boundary.

Online Variable Sized Covering

This paper is the continuation of a previous one devoted to the complexity of standard bases in projective dimension zero (see Giusti, 1989). We improve the upper bound on the maximal degree of elements in a standard basis, with respect to any choice of coordinates and any compatible ordering, given in loc.cit. This bound is sharp, being attained for complete intersections and lexicographic ordering.

This paper is the continuation of a previous one devoted to the complexity of standard bases in projective dimension zero (see Giusti, 1989). We improve the upper bound on the maximal degree of elements in a standard basis, with respect to any choice of coordinates and any compatible ordering, given in loc.cit. This bound is sharp, being attained for complete intersections and lexicographic ordering.