Isoperimetric Type Inequalities Research Articles

Quantum Talagrand, KKL and Friedgut’s Theorems and the Learnability of Quantum Boolean Functions

We extend three related results from the analysis of influences of Boolean functions to the quantum setting, namely the KKL theorem, Friedgut’s Junta theorem and Talagrand’s variance inequality for geometric influences. Our results are derived by a joint use of recently studied hypercontractivity and gradient estimates. These generic tools also allow us to derive generalizations of these results in a general von Neumann algebraic setting beyond the case of the quantum hypercube, including examples in infinite dimensions relevant to quantum information theory such as continuous variables quantum systems. Finally, we comment on the implications of our results as regards to noncommutative extensions of isoperimetric type inequalities, quantum circuit complexity lower bounds and the learnability of quantum observables.

Isoperimetric Type Inequalities for Mappings Induced by Weighted Laplace Differential Operators

Isoperimetric Type Inequalities for Mappings Induced by Weighted Laplace Differential Operators

Weighted isoperimetric inequalities in warped product manifolds

We prove some sharp isoperimetric type inequalities in warped product manifolds, or more generally, multiply warped product manifolds. We also relate them to inequalities involving the higher order mean-curvature integrals. Applications include some sharp eigenvalue estimates, Pólya-Szegö inequality, Faber-Krahn inequality, Sobolev inequality and some sharp geometric inequalities in some warped product spaces.

Sobolev inequality and isoperimetric inequality for submanifolds in a smooth metric measure space

Brendle recently proved a sharp Sobolev inequality and logarithmic Sobolev inequality for submanifolds in Euclidean space. From the sharp Sobolev inequality, he achieved a breakthrough in the conjecture of isoperimetric inequality for minimal submanifolds. In this paper, we extend Brendle’s results to submanifolds in a smooth metric measure space. As an application, we prove some new isoperimetric-type inequalities in some smooth metric measure spaces. For example, we obtain a new isoperimetric-type inequality for self-expander.

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.

A new isoperimetric-type inequality and its stability

A new isoperimetric-type inequality and its stability

On torsional rigidity and ground-state energy of compact quantum graphs

We develop the theory of torsional rigidity—a quantity routinely considered for Dirichlet Laplacians on bounded planar domains—for Laplacians on metric graphs with at least one Dirichlet vertex. Using a variational characterization that goes back to Pólya, we develop surgical principles that, in turn, allow us to prove isoperimetric-type inequalities: we can hence compare the torsional rigidity of general metric graphs with that of intervals of the same total length. In the spirit of the Kohler-Jobin inequality, we also derive sharp bounds on the ground-state energy of a quantum graph in terms of its torsional rigidity: this is particularly attractive since computing the torsional rigidity reduces to inverting a matrix whose size is the number of the graph’s vertices and is, thus, much easier than computing eigenvalues.

General measure extensions of projection bodies

The inequalities of Petty and Zhang are affine isoperimetric-type inequalities providing sharp bounds for $\text{vol}^{n-1}_{n}(K)\text{vol}_n(\Pi^\circ K),$ where $\Pi K$ is a projection body of a convex body $K$. In this paper, we present a number of generalizations of Zhang's inequality to the setting of arbitrary measures. In addition, we introduce extensions of the projection body operator $\Pi$ to the setting of arbitrary measures and functions, while providing associated inequalities for this operator; in particular, Zhang-type inequalities. Throughout, we apply shown results to the standard Gaussian measure.

Some sharp isoperimetric-type inequalities on Riemannian manifolds

We prove some sharp isoperimetric type inequalities for domains with smooth boundary on Riemannian manifolds. For example, using generalized convexity, we show that among all domains with a lower bound l for the cut distance and Ricci curvature lower bound (n−1)k, the geodesic ball of radius l in the space form of curvature k has the largest area-to-volume ratio. A similar but reversed inequality holds if we replace a lower bound on the cut distance by a lower bound of the mean curvature. As an application we show that C2 isoperimetric domains in standard space forms are balls. We also prove another isoperimetric inequality involving the extrinsic radius of a domain when the curvature of the ambient space is bounded above. We then extend it to an inequality which involves the Hausdorff measure of the cut locus.

Isobarycentric Inequalities

Abstract We prove the following isoperimetric-type inequality: Given a finite absolutely continuous Borel measure on ${\mathbb {R}}^n$, half-spaces have maximal measure among all subsets with prescribed barycenter. As a consequence, we make progress towards a solution to a problem of Henk and Pollehn, which is equivalent to a log-Minkowski inequality for a parallelotope and a centered convex body. Our probabilistic approach to the problem also gives rise to several inequalities and conjectures concerning the truncated mean of certain log-concave random variables.

A reverse quantitative isoperimetric type inequality for the Dirichlet Laplacian

A stability result in terms of the perimeter is obtained for the first Dirichlet eigenvalue of the Laplacian operator. In particular, we prove that, once we fix the dimension $n\geq2$, there exists a constant $c>0$, depending only on $n$, such that, for every $\Omega\subset\mathbb{R}^n$ open, bounded, and convex set with volume equal to the volume of a ball $B$ with radius $1$, it holds that $\lambda\_1(\Omega)-\lambda\_1(B)\geq c(P(\Omega)-P(B))^2$, where $\lambda\_1(\cdot)$ denotes the first Dirichlet eigenvalue of a set and $P(\cdot)$ its perimeter. The heart of the present paper is a sharp estimate of the Fraenkel asymmetry in terms of the perimeter.

Necessary and sufficient conditions to Bernstein theorem of a Hessian equation

The Hessian quotient equations S k , l ( D 2 u ) ≡ S k ( D 2 u ) S l ( D 2 u ) = 1 , ∀ x ∈ R n \begin{equation} S_{k,l}(D^2u)\equiv \frac {S_k(D^2u)}{S_l(D^2u)}=1, \ \ \forall x\in {\mathbb {R}}^n \end{equation} were studied for k − k- th symmetric elementary function S k ( D 2 u ) S_k(D^2u) of eigenvalues λ ( D 2 u ) \lambda (D^2u) of the Hessian matrix D 2 u D^2u , where 0 ≤ l > k ≤ n 0\leq l>k\leq n . For l = 0 l=0 , (0.1) is reduced to a k − k- Hessian equation S k ( D 2 u ) = 1 , ∀ x ∈ R n . \begin{equation} S_k(D^2u)=1, \ \ \forall x\in {\mathbb {R}}^n. \end{equation} Two quadratic growth conditions were found by Bao-Cheng-Guan-Ji [American J. Math. 125 (2013), pp. 301–316] ensuring the Bernstein properties of (0.1) and (0.2) respectively. In this paper, we will drop the point wise quadratic growth condition of Bao-Cheng-Guan-Ji and prove three necessary and sufficient conditions to Bernstein property of (0.1) and (0.2), using a reverse isoperimetric type inequality, volume growth or L p L^p -integrability respectively. Our new volume growth or L p − L^p- integrable conditions improve largely various previously known point wise conditions provided Bao et al.; Chen and Xiang [J. Differential Equations 267 (2019), pp. 52027–5219]; Cheng and Yau [Comm. Pure Appl. Math. 39 (1986), pp. 8397–866]; Li, Ren, and Wang [J. Funct. Anal. 270 (2016), pp. 26917–2714]; Yuan [Invent. Math. 150 (2002), pp. 1177–125], etc.

Isoperimetric type inequalities on submanifolds in rotation-symmetric spaces

We study the monotonicity formula involving mean curvature for submanifolds in a rotation-symmetric space $N$. Then we get an isoperimetric type inequality on extrinsic balls of submanifolds in $N$ with a sharp constant.

Foliated corona decompositions

We prove that the $L_4$ norm of the vertical perimeter of any measurable subset of the $3$-dimensional Heisenberg group $\mathbb{H}$ is at most a universal constant multiple of the (Heisenberg) perimeter of the subset. We show that this isoperimetric-type inequality is optimal in the sense that there are sets for which it fails to hold with the $L_4$ norm replaced by the $L_q$ norm for any $q<4$. This is in contrast to the $5$-dimensional setting, where the above result holds with the $L_4$ norm replaced by the $L_2$ norm. The proof of the aforementioned isoperimetric inequality introduces a new structural methodology for understanding the geometry of surfaces in $\mathbb{H}$. In previous work (2017) we showed how to obtain a hierarchical decomposition of Ahlfors-regular surfaces into pieces that are approximately intrinsic Lipschitz graphs. Here we prove that any such graph admits a foliated corona decomposition, which is a family of nested partitions into pieces that are close to ruled surfaces. Apart from the intrinsic geometric and analytic significance of these results, which settle questions posed by Cheeger-Kleiner-Naor (2009) and Lafforgue-Naor (2012), they have several noteworthy implications, including the fact that the $L_1$ distortion of a word-ball of radius $n\ge 2$ in the discrete $3$-dimensional Heisenberg group is bounded above and below by universal constant multiples of $\sqrt[4]{\log n}$; this is in contrast to higher dimensional Heisenberg groups, where our previous work showed that the distortion of a word-ball of radius $n\ge 2$ is of order $\sqrt{\log n}$.

Weighted Isoperimetric Inequalities in Warped Product Manifolds

We prove some isoperimetric type inequalities in warped product manifolds, or more generally, multiply warped product manifolds. We then relate them to inequalities involving the higher order mean-curvature integrals. We also apply our results to obtain sharp eigenvalue estimates and some sharp geometric inequalities in space forms.

Isoperimetric inequalities and geometry of level curves of harmonic functions on smooth and singular surfaces

We investigate the logarithmic convexity of the length of the level curves for harmonic functions on surfaces and related isoperimetric type inequalities. The results deal with smooth surfaces, as well as with singular Alexandrov surfaces (also called surfaces with bounded integral curvature), a class which includes for instance surfaces with conical singularities and surfaces of CAT(0) type. Moreover, we study the geodesic curvature of the level curves and of the steepest descent for harmonic functions on surfaces with non-necessarily constant Gaussian curvature K. Such geodesic curvature functions turn out to satisfy certain Laplace-type equations and inequalities, from which we infer various maximum and minimum principles. The results are complemented by a number of growth estimates for the derivatives L' and L'' of the length of the level curve function L, as well as by examples illustrating the presentation. Our work generalizes some results due to Alessandrini, Longinetti, Talenti, Ma–Zhang and Wang–Wang.

Isoperimetric Type Inequalities and Hypersurface Flows

Isoperimetric Type Inequalities and Hypersurface Flows

Dilation Type Inequalities for Strongly-Convex Sets in Weighted Riemannian Manifolds

Abstract In this paper, we consider a dilation type inequality on a weighted Riemannian manifold, which is classically known as Borell’s lemma in high-dimensional convex geometry. We investigate the dilation type inequality as an isoperimetric type inequality by introducing the dilation profile and estimate it by the one for the corresponding model space under lower weighted Ricci curvature bounds. We also explore functional inequalities derived from the comparison of the dilation profiles under the nonnegative weighted Ricci curvature. In particular, we show several functional inequalities related to various entropies.

The extreme problem for Orlicz and Lq torsional rigidity and their properties

In this paper, the extreme problem for Orlicz and Lq torsional rigidity is discussed. Moreover, we introduce Orlicz and Lq geominimal torsional rigidity, which is defined as being motivated through Orlicz L' mixed torsional rigidity. Also, the invariance of Orlicz and Lq geominimal torsional rigidity under orthogonal matrices is proved, and isoperimetric type inequality and circular type inequality for the torsional rigidity are established as well.

An Isoperimetric-Type Inequality for Spacelike Submanifold in the Minkowski Space

Abstract We prove an isoperimetric-type inequality for maximal, spacelike submanifold in the Minkowski space. The argument is based on the recent work of Brendle.