Benoit Kloeckner
We prove that a plane domain which is almost isoperimetric (with respect to the $L^1$ metric) is close to a square whose sides are parallel to the coordinates axis. Closeness is measured either by $L^\infty$ Haussdorf distance or Fraenkel asymmetry. In the first case, we determine the extremal domains.
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Sharp Lp affine isoperimetric inequalities are established for the entire class of Lp projection bodies and the entire class of Lp centroid bodies. These new inequalities strengthen the Lp Petty projection and the Lp Busemann--Petty centroid inequality.
Using Fourier analysis, we derive Wirtinger-type inequalities of arbitrary high order. As applications, we prove various sharp geometric inequalities for closed curves on the Euclidean plane. In particular, we obtain both sharp lower and upper bounds for the isoperimetric deficit.
Given a compact Riemannian manifold with density $M$ without boundary and the real line $\mathbb{R}$ with constant density, we prove that isoperimetric regions of large volume in $M\times\mathbb{R}$ with the product density are slabs of the form $M\times [a,b]$. We previously prove, as a necessary step, the existence of isoperimetric regions in any manifold of density where a subgroup of the group of transformations preserving weighted perimeter and volume acts cocompactly.
It is well known that isoperimetric inequalities imply in a very general measure-metric-space setting appropriate concentration inequalities. The former bound the boundary measure of sets as a function of their measure, whereas the latter bound the measure of sets separated from sets having half the total measure, as a function of their mutual distance. We show that under a lower bound condition on the Bakry--Émery curvature tensor of a Riemannian manifold equipped with a density, completely general concentration inequalities imply back their isoperimetric counterparts, up to dimension \emph{i…
We obtain symmetrization inequalities on probability metric spaces with convex isoperimetric profile which incorporate in their formulation the isoperimetric estimator and that can be applied to provide a unified treatment of sharp Sobolev-Poincaré and Nash inequalities.
Brunn-Minkowski and Reverse Isoperimetric Inequalities for Dual Quermassintegrals
本文在对偶Brunn-Minkowski理论中建立了两个新几何不等式：证明了Lutwak猜想的原点对称凸体对偶均质积分的Brunn-Minkowski不等式，并推广Ball体积比不等式得出在John位置下立方体最大化对偶均质积分的逆向等周不等式。
In this paper, we develop a basic theory of Orlicz affine and geominimal surface areas for convex and $s$-concave functions. We prove some basic properties for these newly introduced functional affine invariants and establish related functional affine isoperimetric inequalities as well as functional Santaló type inequalities.
We extend the recent $L^{1}$ uncertainty inequalities obtained by Dall'ara-Trevisan to the metric setting. For this purpose we introduce a new class of weights, named *isoperimetric weights*, for which the growth of the measure of their level sets $μ(\{w\leq r\})$ can be controlled by $rI(r),$ where $I$ is the isoperimetric profile of the ambient metric space. We use isoperimetric weights, new *localized Poincaré inequalities*, and interpolation, to prove $L^{p},1\leq p<\infty,$ uncertainty inequalities on metric measure spaces. We give an alternate characterization of the class of isoperimetr…
We introduce the concept of Gaussian integral isoperimetric transference and show how it can be applied to obtain a new class of sharp Sobolev-Poincaré inequalities with constants independent of the dimension. In the special case of $L^{q}$ spaces on the unit $n-$dimensional cube our results extend the recent inequalities that were obtained in \cite{FKS} using extrapolation.
By adapting some ideas of M. Ledoux \cite{ledoux2}, \cite{ledoux-stflour} and \cite{Led} to a sub-Riemannian framework we study Sobolev, Poincaré and isoperimetric inequalities associated to subelliptic diffusion operators that satisfy the generalized curvature dimension inequality that was introduced by F. Baudoin and N. Garofalo in \cite{Bau2}. Our results apply in particular on all CR Sasakian manifolds whose horizontal Webster-Tanaka-Ricci curvature is non negative, all Carnot groups with step two, and wide subclasses of principal bundles over Riemannian manifolds whose Ricci curvature is …
This paper has been withdrawn by the author
Optimal higher-order Sobolev type embeddings are shown to follow via isoperimetric inequalities. This establishes a higher-order analogue of a well-known link between first-order Sobolev embeddings and isoperimetric inequalities. Sobolev type inequalities of any order, involving arbitrary rearrangement-invariant norms, on open sets in $\rn$, possibly endowed with a measure density, are reduced to much simpler one-dimensional inequalities for suitable integral operators depending on the isoperimetric function of the relevant sets. As a consequence, the optimal target space in the relevant Sobol…
We prove that the results regarding the Isoperimetric inequality and Cheeger constant formulated in terms of the Minkowski content, obtained by the authors in previous papers in the framework of essentially non-branching metric measure spaces verifying the local curvature dimension condition, also hold in the stronger formulation in terms of the perimeter.
We prove a family of $L^p$ uncertainty inequalities on fairly general groups and homogeneous spaces, both in the smooth and in the discrete setting. The crucial point is the proof of the $L^1$ endpoint, which is derived from a general weak isoperimetric inequality.
Quantitative isoperimetric inequalities in capillarity problems and cones in strong and barycentric forms
本文针对欧氏半空间中的经典毛细泛函和凸锥中的相对周长，建立了强形式与重心形式的尖锐定量等周不等式，证明等周亏量同时控制 Fraenkel 不对称性和法向量振荡不对称性，并显式给出计算中心。
It is shown that every even, zonal measure on the Euclidean unit sphere gives rise to an isoperimetric inequality for sets of finite perimeter which directly implies the classical Euclidean isoperimetric inequality. The strongest member of this large family of inequalities is shown to be the only affine invariant one among them - the Petty projection inequality. As an application, a family of sharp Sobolev inequalities for functions of bounded variation is obtained, each of which is stronger than the classical Sobolev inequality. Moreover, corresponding families of Lp isoperimetric and Sobolev…
We obtain new sharp isoperimetric inequalities on a Riemannian manifold equipped with a probability measure, whose generalized Ricci curvature is bounded from below (possibly negatively), and generalized dimension and diameter of the convex support are bounded from above (possibly infinitely). Our inequalities are sharp for sets of any given measure and with respect to all parameters (curvature, dimension and diameter). Moreover, for each choice of parameters, we identify the model spaces which are extremal for the isoperimetric problem. In particular, we recover the Gromov--Lévy and Bakry--Le…
We introduce a concept of isoperimetric dimension for magnetic graphs, that is, graphs where every edge is assigned a complex number of modulus one. In analogy with the classical case, we show that isoperimetric inequalities imply Sobolev inequalities on such graphs. As a first application, we show that the signed Cheeger constant behaves additively with respect to Cartesian products of graphs. Using heat kernel techniques, we also give lower bounds for the eigenvalues of the discrete magnetic Laplacian.
In this paper we review our previous isoperimetric results for the logarithmic potential and Newton potential operators. The main reason why the results are useful, beyond the intrinsic interest of geometric extremum problems, is that they produce a priori bounds for spectral invariants of operators on arbitrary domains. We demonstrate these in explicit examples.
Isoperimetric inequalities in high-dimensional convex sets
本讲义笔记围绕Bourgain切片问题及Kannan、Lovasz与Simonovits提出的KLS等周猜想的近期研究进展展开阐述。
We prove new $L_p$ affine isoperimetric inequalities for all $ p \in [-\infty,1)$. We establish, for all $p\neq -n$, a duality formula which shows that $L_p$ affine surface area of a convex body $K$ equals $L_\frac{n^2}{p}$ affine surface area of the polar body $K^\circ$.
The Blaschke Santaló inequality and the $L_p$ affine isoperimetric inequalities are major inequalities in convex geometry and they have a wide range of applications. Functional versions of the Blaschke Santaló inequality have been established over the years through many contributions. More recently and ongoing, such functional versions have been established for the $L_p$ affine isoperimetric inequalities as well. These functional versions involve notions from information theory, like entropy and divergence. We list stability versions for the geometric inequalities as well as for their function…
In recent years, it has been shown that some classical inequalities follow from a local stochastic dominance for naturally associated random polytopes. We strengthen planar isoperimetric inequalities by attaching a stochastic model to some classical inequalities, such as Mahler's Theorem, and a reverse Lutwak-Zhang inequality, the polar for $L_p$ centroid bodies. In particular, we obtain the dual counterpart to a result of Bisztriczky-Böröczky.
In this paper we prove an isoperimetric inequality of euclidean type for complete metric spaces admitting a cone-type inequality. These include all Banach spaces and all complete, simply-connected metric spaces of non-positive curvature in the sense of Alexandrov or, more generally, of Busemann. The main theorem generalizes results of Gromov and Ambrosio-Kirchheim.
Let $(E,\F,μ)$ be a $\si$-finite measure space. For a non-negative symmetric measure $J(\d x, \d y):=J(x,y) \,μ(\d x)\,μ(\d y)$ on $E\times E,$ consider the quadratic form $$\E(f,f):= \frac{1}{2}\int_{E\times E} (f(x)-f(y))^2 \, J(\d x,\d y)$$ in $L^2(μ)$. We characterize the relationship between the isoperimetric inequality and the super Poincaré inequality associated with $\E$. In particular, sharp Orlicz-Sobolev type and Poincaré type isoperimetric inequalities are derived for stable-like Dirichlet forms on $\R^n$, which include the existing fractional isoperimetric inequality as a special …
This is a continuation of our previous work 0712.4092. It is well known that various isoperimetric inequalities imply their functional ``counterparts'', but in general this is not an equivalence. We show that under certain convexity assumptions (e.g. for log-concave probability measures in Euclidean space), the latter implication can in fact be reversed for very general inequalities, generalizing a reverse form of Cheeger's inequality due to Buser and Ledoux. We develop a coherent single framework for passing between isoperimetric inequalities, Orlicz-Sobolev functional inequalities and capaci…
We solve a class of isoperimetric problems on $\mathbb{R}^N $ with respect to weights that are powers of the distance to the origin. For instance we show that if $k\in [0,1]$, then among all smooth sets $Ω$ in $\mathbb{R} ^N$ with fixed Lebesgue measure, $\int_{\partial Ω} |x|^k \, \mathscr{H}_{N-1} (dx)$ achieves its minimum for a ball centered at the origin. Our results also imply a weighted Polya-Szëgo principle. In turn, we establish radiality of optimizers in some Caffarelli-Kohn-Nirenberg inequalities, and we obtain sharp bounds for eigenvalues of some nonlinear problems.
We prove upper and lower bounds of the heat kernel for the operator $Δ-\nabla (\frac{1}{|x|^α})\cdot \nabla $ in $\mathbb{R}^{n}\setminus\{0} $ where $α>0$. We obtain these bounds from an isoperimetric inequality for a measure $\mathrm{e}^{-\frac{1}{|x|^α}}dx$ on $\mathbb{R}^{n}\setminus \{0\}$. The latter amounts to a certain functional isoperimetric inequality for the radial part of this measure.
In this paper we prove that the disc is a maximiser of the Schatten $p$-norm of the logarithmic potential operator among all domains of a given measure in $\mathbb R^{2}$, for all even integers $2\leq p<\infty$. We also show that the equilateral triangle has the largest Schatten $p$-norm among all triangles of a given area. For the logarithmic potential operator on bounded open or triangular domains, we also obtain analogies of the Rayleigh-Faber-Krahn or P{ó}lya inequalities, respectively. The logarithmic potential operator can be related to a nonlocal boundary value problem for the Laplacian…
In this paper we adapt the well-estabilished $Γ$-calculus techniques to the context of $RCD(K,\infty)$ spaces, proving Bobkov's local isoperimetric inequality and, when $K$ is positive, the Gaussian isoperimetric inequality in this class of spaces. The proof relies on the measure-valued $Γ_2$ operator introduced by Savaré.
Optimal subelliptic super-Poincaré and isoperimetric inequalities on stratified Lie groups
本文在分层李群及 Grushin、Heisenberg-Greiner 等次椭圆情形中，对指数幂型概率测度证明了 $q\in[1,2]$ 的超庞加莱不等式，并得到一般最优等周不等式。
Firstly, we derive in dimension one a new covariance inequality of $L_{1}-L_{\infty}$ type that characterizes the isoperimetric constant as the best constant achieving the inequality. Secondly, we generalize our result to $L_{p}-L_{q}$ bounds for the covariance. Consequently, we recover Cheeger's inequality without using the co-area formula. We also prove a generalized weighted Hardy type inequality that is needed to derive our covariance inequalities and that is of independent interest. Finally, we explore some consequences of our covariance inequalities for $L_{p}$-Poincaré inequalities and …
Isoperimetric and geometric inequalities in quantitative form: Stein's method approach
该研究通过引入带斜边界条件的椭圆型PDE解决边界项处理难题，将Stein方法几何化，得到Brock-Weinstock不等式、Steklov特征值约束下等周不等式及加权与未加权周长组合的稳定性结果，均基于α-Zolotarev距离表述。
The $k$-th symmetric product of a graph $G$ with vertex set $V$ with edge set $E$ is a graph with vertices as $k$-sets of $V$, where two $k$-sets are connected by an edge if and only if their symmetric difference is an edge in $E$. Using the isoperimetric properties of the vertex-induced subgraphs of $G$ and Sobolev inequalities on graphs, we obtain various edge-isoperimetric inequalities pertaining to the symmetric product of certain families of finite and infinite graphs.
Affine Fractional Sobolev and Isoperimetric Inequalities
本文建立 $\mathbb R^n$ 上的尖锐仿射分数阶 Sobolev 不等式，强化并推出 Almgren–Lieb 不等式，极限下蕴含 Zhang 的仿射 Sobolev 不等式，同时获得分数阶 Petty 投影不等式并证明径向平均体猜想。
We study functional inequalities for Markov chains on discrete spaces with entropic Ricci curvature bounded from below. Our main results are that when curvature is non-negative, but not necessarily positive, the spectral gap, the Cheeger isoperimetric constant and the modified logarithmic Sobolev constant of the chain can be bounded from below by a constant that only depends on the diameter of the space, with respect to a suitable metric. These estimates are discrete analogues of classical results of Riemannian geometry obtained by Li and Yau, Buser and Wang.
In this note we prove that the ball is a maximiser of some Schatten $p$-norms of the Riesz potential operators among all domains of a given measure in $\mathbb R^{d}$. In particular, the result is valid for the polyharmonic Newton potential operator, which is related to a nonlocal boundary value problem for the poly-Laplacian extending the one considered by M. Kac in the case of the Laplacian, so we obtain and isoperimetric inequalities for its eigenvalues as well, namely, analogues of Rayleigh-Faber-Krahn and Hong-Krahn-Szegö inequalities.
Various properties of isoperimetric, functional, Transport-Entropy and concentration inequalities are studied on a Riemannian manifold equipped with a measure, whose generalized Ricci curvature is bounded from below. First, stability of these inequalities with respect to perturbation of the measure is obtained. The extent of the perturbation is measured using several different distances between perturbed and original measure, such as a one-sided $L^\infty$ bound on the ratio between their densities, Wasserstein distances, and Kullback--Leibler divergence. In particular, an extension of the Hol…
The concentration of measure prenomenon roughly states that, if a set $A$ in a product $Ω^N$ of probability spaces has measure at least one half, ``most'' of the points of $Ω^N$ are ``close'' to $A$. We proceed to a systematic exploration of this phenomenon. The meaning of the word ``most'' is made rigorous by isoperimetric-type inequalities that bound the measure of the exceptional sets. The meaning of the work ``close'' is defined in three main ways, each of them giving rise to related, but different inequalities. The inequalities are all proved through a common scheme of proof. Remarkably, …
Anisotropic symmetrization, convex bodies, and isoperimetric inequalities
本文利用各向异性等周不等式、Sobolev 函数精细性质及 Brunn-Minkowski 理论，给出各向异性 Pólya-Szegö 型不等式无需逼近论证的新证明，并实现对极值函数的刻画。
On isoperimetric local-Bollobás-Thomason inequalities
该研究证明了关于凸体的等周型不等式，通过构造合适的Hanner多面体来保证投影体积的下界。
Star-shaped bodies are an important nonconvex generalization of convex bodies (e.g., linear programming with violations). Here we present an efficient algorithm for sampling a given star-shaped body. The complexity of the algorithm grows polynomially in the dimension and inverse polynomially in the fraction of the volume taken up by the kernel of the star-shaped body. The analysis is based on a new isoperimetric inequality. Our main technical contribution is a tool for proving such inequalities when the domain is not convex. As a consequence, we obtain a polynomial algorithm for computing the …
On stochastic forms of functional isoperimetric inequalities
本文研究了p-凹函数类中的功能等周不等式的随机形式，推导出Sobolev型不等式，并验证了Zhang的仿射Sobolev不等式在随机模型中的期望成立，同时恢复了几何和确定性结果。
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