Isoperimetric Research Articles

Core Shells and Double Bubbles in a Weighted Nonlocal Isoperimetric Problem

Core Shells and Double Bubbles in a Weighted Nonlocal Isoperimetric Problem

Reflect-Push Methods Part I: Two Dimensional Techniques

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.

Isoperimetric inequalities for real‐valued functions with applications to monotonicity testing

AbstractWe generalize the celebrated isoperimetric inequality of Khot, Minzer, and Safra (SICOMP 2018) for Boolean functions to the case of real‐valued functions . Our main tool in the proof of the generalized inequality is a new Boolean decomposition that represents every real‐valued function over an arbitrary partially ordered domain as a collection of Boolean functions over the same domain, roughly capturing the distance of to monotonicity and the structure of violations of to monotonicity. We apply our generalized isoperimetric inequality to improve algorithms for testing monotonicity and approximating the distance to monotonicity for real‐valued functions. Our tester for monotonicity has query complexity , where is the size of the image of the input function. (The best previously known tester makes queries, as shown by Chakrabarty and Seshadhri (STOC 2013).) Our tester is nonadaptive and has 1‐sided error. We prove a matching lower bound for nonadaptive, 1‐sided error testers for monotonicity. We also show that the distance to monotonicity of real‐valued functions that are ‐far from monotone can be approximated nonadaptively within a factor of with query complexity polynomial in and the dimension . This query complexity is nearly optimal for nonadaptive algorithms even for the special case of Boolean functions (The best previously known distance approximation algorithm for real‐valued functions, by Fattal and Ron (TALG 2010) achieves ‐approximation.).

Optimization of layout for embedding half hypercube into conventional tree architectures

SummaryEmbedding a graph into another graph can be utilized for structural simulation, processor allocation, and algorithm porting in the field of parallel architecture. This has the potential to enhance the physical layout of network‐on‐chip (NoC) devices as well as to investigate their virtualization possibilities. Layout is one of the many indicators of graph embedding. An optimal layout in NoC design can result in a decreased wiring area and cost, as well as the reduction in communication delay between parallel processing components. In this work, the guest graph is the half hypercube, which possess efficient routing, fault tolerance, and Hamiltonian features. The edge isoperimetric problems is solved for the half hypercube using a novel technique called the isochronal ordering. The host graphs considered in our work are complete binary trees, ‐rooted complete binary trees, and other few predominantly investigated tree based architectures.

Stability for the complete intersection theorem, and the forbidden intersection problem of Erdős and Sós

A family \mathcal{F} of sets is said to be t-intersecting if |A \cap B| \geq t for any A,B \in \mathcal{F} . The seminal Complete Intersection Theorem of Ahlswede and Khachatrian (1997) gives the maximal size f(n,k,t) of a t -intersecting family of k -element subsets of [n]=\{1,\ldots,n\} , together with a characterisation of the extremal families, solving a longstanding problem of Frankl. The forbidden intersection problem , posed by Erdős and Sós in 1971, asks for a determination of the maximal size g(n,k,t) of a family \mathcal{F} of k -element subsets of [n] such that |A \cap B| \neq t-1 for any A,B \in \mathcal{F} . In this paper, we show that for any fixed t \in \mathbb{N} , if o(n) \leq k \leq n/2-o(n) , then g(n,k,t)=f(n,k,t) . In combination with prior results, this solves the problem of Erdős and Sós for any constant t , except for the ranges n/2-o(n) < k < n/2+t/2 and k < 2t . One key ingredient of the proof is the following sharp ‘stability’ result for the Complete Intersection Theorem: if k/n is bounded away from 0 and 1/2 , and \mathcal{F} is a t -intersecting family of k -element subsets of [n] such that |\mathcal{F}| \geq f(n,k,t) - O(\binom{n-d}{k}) , then there exists a family \mathcal{G} such that \mathcal{G} is extremal for the Complete Intersection Theorem, and |\mathcal{F} \setminus \mathcal{G}| = O(\binom{n-d}{k-d}) . This proves a conjecture of Friedgut (2008). We prove the result by combining classical ‘shifting’ arguments with a ‘bootstrapping’ method based upon an isoperimetric inequality. Another key ingredient is a ‘weak regularity lemma’ for families of k -element subsets of [n] , where k/n is bounded away from 0 and 1. This states that any such family \mathcal{F} is approximately contained within a ‘junta’ such that the restriction of \mathcal{F} to each subcube determined by the junta is ‘pseudorandom’ in a certain sense.

An isoperimetric problem with two distinct solutions

In this paper we prove that among all convex domains of the plane with two axes of symmetry, the maximizer of the first non-trivial Neumann eigenvalue μ 1 \mu _1 with perimeter constraint is achieved by the square and the equilateral triangle. Part of the result follows from a new general bound on μ 1 \mu _1 involving the minimal width over the area. Our main result partially answers to a question addressed in 2009 by R. S. Laugesen, I. Polterovich, and B. A. Siudeja.

Private Convex Optimization via Exponential Mechanism

In this paper, we study private optimization problems for non-smooth convex functions $F(x)=\mathbb{E}_i f_i(x)$ on $\mathbb{R}^d$.We show that modifying the exponential mechanism by adding an $\ell_2^2$ regularizer to $F(x)$ and sampling from $\pi(x)\propto \exp(-k(F(x)+\mu\|x\|_2^2/2))$ recovers both the known optimal empirical risk and population loss under $(\eps,\delta)$-DP. Furthermore, we show how to implement this mechanism using $\widetilde{O}(n \min(d, n))$ queries to $f_i(x)$ for the DP-SCO where $n$ is the number of samples/users and $d$ is the ambient dimension.We also give a (nearly) matching lower bound $\widetilde{\Omega}(n \min(d, n))$ on the number of evaluation queries.
 Our results utilize the following tools that are of independent interest:\begin{itemize}\item We prove Gaussian Differential Privacy (GDP) of the exponential mechanism if the loss function is strongly convex and the perturbation is Lipschitz. Our privacy bound is \emph{optimal} as it includes the privacy of Gaussian mechanism as a special case and is proved using the isoperimetric inequality for strongly log-concave measures.\item We show how to sample from $\exp(-F(x)-\mu \|x\|^2_2/2)$ for $G$-Lipschitz $F$ with $\eta$ error in total variation (TV) distance using $\widetilde{O}((G^2/\mu) \log^2(d/\eta))$ unbiased queries to $F(x)$. This is the first sampler whose query complexity has \emph{polylogarithmic dependence} on both dimension $d$ and accuracy $\eta$.\end{itemize}

Isoperimetric inequalities and regularity of A-harmonic functions on surfaces

We investigate the logarithmic and power-type convexity of the length of the level curves for a-harmonic functions on smooth surfaces and related isoperimetric inequalities. In particular, our analysis covers the p-harmonic and the minimal surface equations. As an auxiliary result, we obtain higher Sobolev regularity properties of the solutions, including the W2,2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$W^{2,2}$$\\end{document} regularity. The results are complemented by a number of estimates for the derivatives L′\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L'$$\\end{document} and L′′\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L''$$\\end{document} of the length of the level curve function L, as well as by examples illustrating the presentation. Our work generalizes results due to Alessandrini, Longinetti, Talenti and Lewis in the Euclidean setting, as well as a recent article of ours devoted to the harmonic case on surfaces.

Parabolicity on Graphs

Large scale properties of Riemannian manifolds, in particular, those properties preserved by quasi-isometries, can be studied using discrete structures which approximate the manifolds. In a sequence of papers, M. Kanai proved that, under mild conditions, many properties are preserved by a certain (quasi-isometric) graph approximation of a manifold. One of these properties is p-parabolicity. A manifold M (respectively, a graph G) is said to be p-parabolic if all positive p-superharmonic functions on M (resp. G) are constant. This is equivalent to not having p-Green’s function (i.e. a positive fundamental solution of the p-Laplace-Beltrami operator). Herein we study directly the p-parabolicity on graphs. We obtain some characterizations in terms of graph decompositions. Also, we give necessary and sufficient conditions for a uniform hyperbolic graph to be p-parabolic in terms of its boundary at infinity. Finally, we prove that if a uniform hyperbolic graph satisfies the (Cheeger) isoperimetric inequality, then it is non-p-parabolic for every 1<p<∞\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$1<p<\\infty $$\\end{document}.

Applications of possibly hidden symmetry to Steklov and mixed Steklov problems on surfaces

We consider three different questions related to the Steklov and mixed Steklov problems on surfaces. These questions are connected by the techniques that we use to study them, which exploit symmetry in various ways even though the surfaces we study do not necessarily have inherent symmetry. In the spirit of the celebrated Hersch-Payne-Schiffer and Weinstock inequalities for Steklov eigenvalues, we obtain a sharp isoperimetric inequality for the mixed Steklov eigenvalues considering the interplay between the eigenvalues of the mixed Steklov-Neumann and Steklov-Dirichlet eigenvalues. In 1980, Bandle showed that the unit disk maximizes the kth nonzero normalized Steklov eigenvalue on simply connected domains with rotational symmetry of order p when k≤p−1. We discuss whether the disk remains the maximizer in the class of simply connected rotationally symmetric domains when k≥p. In particular, we show that as k→∞, the upper bound converges to the Hersch-Payne-Schiffer upper bound. We give full asymptotics for mixed Steklov problems on arbitrary surfaces, assuming some conditions at the meeting points of the Steklov boundary with the Dirichlet or Neumann boundary.

Isoperimetric inequalities in nonlocal diffusion problems with integrable kernel

We deduce isoperimetric estimates for solutions of linear stationary and evolution problems. Our main result establishes the comparison in norm between the solution of a problem and its symmetric version when nonlocal diffusion defined through integrable kernels is replacing the usual local diffusion defined by a second order differential operator. Since an appropriate kernel rescaling allows to define a sequence of solutions of the nonlocal diffusion problems converging to their local diffusion counterparts, we also find the corresponding isoperimetric inequalities for the latter, i.e. we prove the classical Talenti's theorem. The novelty of our approach is that we replace the measure geometric tools employed in Talenti's proof, such as the geometric isoperimetric inequality or the coarea formula, by the Riesz's rearrangement inequality. Thus, in addition to providing a proof for the nonlocal diffusion case, our technique also introduces an alternative proof to Talenti's theorem.

Hypercontractivity on the symmetric group

Abstract The hypercontractive inequality is a fundamental result in analysis, with many applications throughout discrete mathematics, theoretical computer science, combinatorics and more. So far, variants of this inequality have been proved mainly for product spaces, which raises the question of whether analogous results hold over non-product domains. We consider the symmetric group, $S_n$ , one of the most basic non-product domains, and establish hypercontractive inequalities on it. Our inequalities are most effective for the class of global functions on $S_n$ , which are functions whose $2$ -norm remains small when restricting $O(1)$ coordinates of the input, and assert that low-degree, global functions have small q-norms, for $q&gt;2$ . As applications, we show the following: 1. An analog of the level-d inequality on the hypercube, asserting that the mass of a global function on low degrees is very small. We also show how to use this inequality to bound the size of global, product-free sets in the alternating group $A_n$ . 2. Isoperimetric inequalities on the transposition Cayley graph of $S_n$ for global functions that are analogous to the KKL theorem and to the small-set expansion property in the Boolean hypercube. 3. Hypercontractive inequalities on the multi-slice and stability versions of the Kruskal–Katona Theorem in some regimes of parameters.

On the quantitative isoperimetric inequality in the plane with the barycentric asymmetry

On the quantitative isoperimetric inequality in the plane with the barycentric asymmetry

The upper bound of the harmonic mean of the Steklov eigenvalues in curved spaces

AbstractConsidering an ‐dimensional Riemannian manifold whose sectional curvature is bounded above by and Ricci curvature is bounded below by , we obtain an upper bound of the harmonic mean of the first nonzero Steklov eigenvalues for domains contained in . This can be viewed as certain isoperimetric inequality and generalizes the result on comparing the first nonzero Steklov eigenvalues for such domains (Li–Wang–Wu, 2020, arXiv:2003.03093).

Метод синтеза нелинейных квазиоптимальных законов управления многомерными системами на основе условия максимума функции обобщенной мощности и анализа пучка квадратичных форм

The synthesis of the control law structure for multidimensional dynamic system according to a quadratic criterion based on the reduction of the Lagrange problem to an isoperimetric problem was carried out taking into account the analysis of a quadratic forms bundle, which ensures the fulfillment of transversality conditions. This makes it possible to take into account the dynamic properties of the controlled system when constructing quasi-optimal control. Indeterminate coupling multipliers of the second kind can be established as a result of using the principle of releasability, taking into account the property of Poisson brackets based on numerical modeling. The analysis of the modeling results of the control process of a dual pendulum shows that the developed quasi-optimal control law of a nonlinear dynamic system with several degrees of freedom results in a gain in terms of speed when improving the value of the quadratic criterion in comparison with the known control based on the principle of decomposition and the game approach.

A unified approach to higher order discrete and smooth isoperimetric inequalities

AbstractWe present a unified approach to derive sharp isoperimetric‐type inequalities of arbitrary high order. In particular, we obtain (i) sharp high‐order discrete polygonal isoperimetric‐type inequalities, (ii) sharp high‐order isoperimetric‐type inequalities for smooth curves with both upper and lower bounds for the isoperimetric deficit, and (iii) sharp higher order Chernoff‐type inequalities involving a generalized width function and higher order locus of curvature centers. Our approach involves obtaining higher order discrete or smooth Wirtinger inequalities via Fourier analysis, by examining a family of linear operators. The key to our approach is identifying the appropriate linear operator and translating the analytic inequalities into geometric ones.

Anchored heat kernel upper bounds on graphs with unbounded geometry and anti-trees

We derive Gaussian heat kernel bounds on graphs with respect to a fixed origin for large times under the assumption of a Sobolev inequality and volume doubling on large balls. The main result is then applied to anti-trees with unbounded vertex degree, yielding Gaussian upper bounds for this class of graphs for the first time. In order to prove this, we show that isoperimetric estimates with respect to intrinsic metrics yield Sobolev inequalities. Finally, we prove that anti-trees are Ahlfors regular and that they satisfy an isoperimetric inequality of a larger dimension.

Affine isoperimetric inequalities on flag manifolds

Abstract Building on work of Furstenberg and Tzkoni, we introduce $\mathbf {r}$ -flag affine quermassintegrals and their dual versions. These quantities generalize affine and dual affine quermassintegrals as averages on flag manifolds (where the Grassmannian can be considered as a special case). We establish affine and linear invariance properties and extend fundamental results to this new setting. In particular, we prove several affine isoperimetric inequalities from convex geometry and their approximate reverse forms. We also introduce functional forms of these quantities and establish corresponding inequalities.

Weak commutativity, virtually nilpotent groups, and Dehn functions

The group \mathfrak{X}(G) is obtained from G\ast G by forcing each element g in the first free factor to commute with the copy of g in the second free factor. We make significant additions to the list of properties that the functor \mathfrak{X} is known to preserve. We also investigate the geometry and complexity of the word problem for \mathfrak{X}(G) . Subtle features of \mathfrak{X}(G) are encoded in a normal abelian subgroup W&lt;\mathfrak{X}(G) that is a module over \mathbb{Z} Q , where Q= H_1(G,\mathbb{Z}) . We establish a structural result for this module and illustrate its utility by proving that \mathfrak{X} preserves virtual nilpotence, the Engel condition, and growth type – polynomial, exponential, or intermediate. We also use it to establish isoperimetric inequalities for \mathfrak{X}(G) when G lies in a class that includes Thompson's group F and all non-fibred Kähler groups. The word problem is soluble in \mathfrak{X}(G) if and only if it is soluble in G . The Dehn function of \mathfrak{X}(G) is bounded below by a cubic polynomial if G maps onto a non-abelian free group.

A Proof of Finite Crystallization via Stratification

We devise a new technique to prove two-dimensional crystallization results in the square lattice for finite particle systems. We apply this strategy to energy minimizers of configurational energies featuring two-body short-ranged particle interactions and three-body angular potentials favoring bond-angles of the square lattice. To each configuration, we associate its bond graph which is then suitably modified by identifying chains of successive atoms. This method, called stratification, reduces the crystallization problem to a simple minimization that corresponds to a proof via slicing of the isoperimetric inequality in ℓ1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\ell ^1$$\\end{document}. As a byproduct, we also prove a fluctuation estimate for minimizers of the configurational energy, known as the n3/4\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$n^{3/4}$$\\end{document}-law.