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Kanai proved the stability under quasi-isometries of numerous global properties (including isoperimetric inequalities) between Riemannian manifolds of bounded geometry. Even though quasi-isometries highly distort local properties, recently it was shown that the injectivity radius is preserved (in some appropriate sense) under these maps between genus zero Riemann surfaces. In the present work, results along these lines are obtained even for infinite genus. As a consequence, the stability of the isoperimetric inequality in this context (without the hypothesis of bounded geometry) is also obtained.

We consider isoperimetric problem with a nonlocal repulsive term given by the Newtonian potential. We prove that regular critical sets of the functional are analytic. This optimal regularity holds also for critical sets of the Ohta–Kawasaki functional. We also prove that when the strength of the nonlocal part is small the ball is the only possible stable critical set.

The main focus of this paper is on hyperbolic properties of multiply connected planar graphs (planar graphs with multiple ends), and in the course we study some problematic phenomena of planar graphs caused by the existence of multiple (or sometimes infinite) ends. Specifically, in the first part of the paper we examine strong isoperimetric inequalities on a multiply connected planar graph G and its dual graph $$G^*$$Gź, and prove that G satisfies a strong isoperimetric inequality if and only if $$G^*$$Gź has the same property, provided that G is either normal or finitely connected and we choose an appropriate notion for strong isoperimetric inequalities. In the second part we study a planar graph G on which negative curvatures uniformly dominate positive curvatures, and give a criterion that guarantees a strong isoperimetric inequality on G. Our criterion is useful in that it can be applied to a graph containing a long and slim subgraph with nonnegative combinatorial curvatures.

International audience; Consider an open domain D on the plane, whose isoperimetric deficit is smaller than 1. This note shows that the difference between the barycenter of D and the barycenter of its boundary is bounded above by a constant times the isoperimetric deficit to the power 1/4. This power can be improved to 1/2, when D is furthermore assumed to be a convex domain, in any Euclidean space of dimension larger than 2. Keywords: Isoperimetric inequality on the plane, isoperimetric deficit, boundary barycenter, convex domains, isoperimetric stability.

A well-known question in differential geometry is to control the constant in isoperimetric inequality by intrinsic curvature conditions. In dimension 2, the constant can be controlled by the integral of the positive part of the Gaussian curvature. In this paper, we showed that on simply connected conformally flat manifolds of higher dimensions, the role of the Gaussian curvature can be replaced by the Branson's Q-curvature. We achieve this by exploring the relationship between A p weights and integrals of the Q-curvature.

In the Heisenberg group H^n, we prove quantitative isoperimetric inequalities for Pansu's spheres, that are known to be isoperimetric under various assumptions. The inequalities are shown for suitably restricted classes of competing sets and the proof relies on the construction of sub-calibrations.

We give an analytic proof of the quantitative isoperimetric inequality in the plane and give an estimation of the upper bound of the constant via maximizing the L ∞ -norm of the gradient of solutions to the Poisson equation.

We study equivalent conditions for the Cheeger isoperimetric constant of Riemannian manifolds to be positive. We first give a proof of Gromov’s assertion for locally symmetric spaces with infinite volume, which states that the existence of a bounded primitive of the Riemannian volume form is equivalent to the positivity of the Cheeger isoperimetric constant. Furthermore, under the assumption of pinched negative sectional curvature, we obtain another equivalent condition in terms of bounded cohomology classes. This generalizes Soma’s result (Duke Math J 88(2):357–370, 1997) for hyperbolic 3-manifolds to \({\mathbb {R}}\)-rank one locally symmetric spaces.

investigate the symmetric mixed isoperimetric deficit Δ2( K 0, K 1) of domains K 0 and K 1 in the Euclidean plane R 2 via the known kinematic formulas of Poincare and Blaschke in integral geometry. Then we obtain symmetric mixed isoperimetric inequality and some Bonnesen-style symmetric mixed inequalities. One of those inequalities strengthens Kotlyars inequality. Finally, we obtain some reverse Bonnesen-style symmetric mixed inequalities that generalize the known Bottemas result.

The paper considers geometric lower bounds on the isoperimetric constant for logarithmically concave probability measures, extending and refining some results by Kannan et al. (Discret Comput Geom 13:541–559, 1995).

We establish the validity of a quantitative isoperimetric inequality in higher codimension. To be precise we show for any closed (n-1)-dimensional manifold Γ in R^{n+k} that the quantitative isoperimetric inequality D(Γ)≥ C_1 d^2(Γ) holds true. Here D(Γ) stands for the isoperimetric deficit of Γ, i.e., the deviation in measure of Γ being a round sphere. Further, d(Γ ) denotes a natural generalization to higher codimension of the Fraenkel asymmetry index of Γ.

In this paper we study the existence and the uniqueness of isoperimetric extremals of rotation on two-dimensional (pseudo-) Riemannian manifolds and on surfaces on Euclidean space. We find the new form of their equations which is easier than results by S. G. Leiko. He introduced the notion of rotary diffeomorphisms. In this paper we propose a new proof of the fundamental equations of rotary mappings.

We prove an Hersch's type isoperimetric inequality for the third positive eigenvalue on $\mathbb S^2$. Our method builds on the theory we developped to construct extremal metrics on Riemannian surfaces in conformal classes for any eigenvalue.

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.

For embedded closed curves with curvature bounded below, we prove an isoperimetric inequality estimating the minimal area bounded by such curves for a fixed perimeter.

We generalize the Poincare limit which asserts that the n -dimensional Gaussian measure is approximated by the projections of the uniform probability measure on the Euclidean sphere of appropriate radius to the first n -coordinates as the dimension diverges to infinity. The generalization is done by replacing the projections with certain maps. Using this generalization, we derive a Gaussian isoperimetric inequality for an absolutely continuous probability measure on Euclidean spaces with respect to the Lebesgue measure, whose density is a radial function.

This paper deals with the eigenvalues of the Neumann Laplacian on simply-connected Lipschitz planar domains with some rotational symmetry. Our aim is to continue the investigations from Enache and Philippin [7] and derive new isoperimetric estimates for eigenvalues of higher order.

In this paper, the Orlicz-affine surface area is introduced. Isoperimetric inequalities for this new afine surface area are established. Mathematics subject classification (2010): Primary 52A20; Secondary 53A15.

In this note we derive a maximum principle for an appropriate functional combination of u ( x ) and | ∇ u | 2 , where u ( x ) is a strictly convex classical solution to a general class of Monge–Ampere equations. This maximum principle is then employed to establish some isoperimetric inequalities of interest in the theory of surfaces of constant Gauss curvature in R N + 1 .

We consider integral functionals of a simply connected domain which depend on the distance to the domain boundary. We prove an isoperimetric inequality generalizing theorems derived by the Schwarz symmetrization method. For Lp-norms of the distance function we prove an analog of the Payne inequality for the torsional rigidity of the domain. In compare with the Payne inequality we find new extremal domains different from a disk.

For a wedge-like membrane, Payne and Weinberger proved in 1960 an isoperimetric inequality for the fundamental eigenvalue which in some cases improves the classical isoperimetric inequality of Faber–Krahn. In this work, we introduce “relative torsional rigidity” for this type of membrane and prove new isoperimetric inequalities in the spirit of Saint-Venant, Polya–Szegő, Payne, Payne–Rayner, Chiti, and Talenti, which link the eigenvalue problem with the boundary value problem in a fundamental way.

In this paper we study the quantitative isoperimetric inequality in the plane. We prove the existence of a set $\Omega$, different from a ball, which minimizes the ratio $\delta(\Omega)/\lambda^2(\Omega)$, where $\delta$ is the isoperimetric deficit and $\lambda$ the Fraenkel asymmetry, giving a new proof ofthe quantitative isoperimetric inequality. Some new properties of the optimal set are also shown., Comment: Equipe {\'e}quations aux d{\'e}riv{\'e}es partielles

We discuss isoperimetric inequalities for convex sets. These include the classical isoperimetric inequality and that of Brunn-Minkowski, Blaschke-Santalo, Busemann-Petty and their various extensions. We show that many such inequalities admit stronger randomized forms in the following sense: for natural families of associated random convex sets one has stochastic dominance for various functionals such as volume, surface area, mean width and others. By laws of large numbers, these randomized versions recover the classical inequalities. We give an overview of when such stochastic dominance arises and its applications in convex geometry and probability.

Taxicab plane geometry and Cinese-Checker plane geometry are non-Euclidean and more practical notion than Euclidean geometry in the real world. The -distance is a generalization of the Taxicab distance and Chinese-Checker distance. It was first introduced by Songlin Tian in 2005, and generalized to n-dimensional space by Ozcan Gelisgen in 2006. In this paper, we studied the isoperimetric inequality in -plane.

We prove existence of regions minimizing perimeter under a volume constraint in contact sub-Riemannian manifolds such that their quotient by the group of contact transformations preserving the sub-Riemannian metric is compact., 24 pages, no figures

We give an example of a smooth surface of revolution for which all circles about the origin are strictly stable for fixed area but small isoperimetric regions are nearly round discs away from the origin.

We prove that a proper geodesic metric space has non-positive curvature in the sense of Alexandrov if and only if it satisfies the Euclidean isoperimetric inequality for curves. Our result extends to non-geodesic spaces and non-zero curvature bounds.

We describe a general technique for embedding certain amalgamated products into direct products. This technique provides us with a way of constructing a host of finitely presented subgroups of automatic groups which are not even asynchronously automatic. We can also arrange that such subgroups satisfy, at best, an exponential isoperimetric inequality., Comment: DVI and Post-Script files only. To appear in J. London Math. Soc

It is shown that D. Cohen's inequality bounding the isoperimetric function of a group by the double exponential of its isodiametric function is valid in the more general context of locally finite simply connected complexes. It is shown that in this context this bound is ‘best possible’. Also studied are second-dimensional isoperimetric functions for groups and complexes. It is shown that the second-dimensional isoperimetric function of a group is bounded by a recursive function. By a similar argument it is shown that the area distortion of a finitely presented subgroup of a finitely presented group is recursive. Cohen's inequality is extended to second-dimensional isoperimetric and isodiametric functions of 2-connected simplicial complexes.

We consider three generalizations of the isoperimetric problem to higher codimension and provide results on equilibrium, stability, and minimization., 13 pages, 1 figure; v2: Minor revision to appear in Manuscripta Mathematica

We prove sharp quantitative isoperimetric inequalities for John domains in R n . We show that the Bonnesen-style inequalities hold true in R n under the John domain assumption which rules out cusps. Our main tool is a proof of the isoperimetric inequality for symmetric domains which gives an explicit estimate for the isoperimetric deficit. We use the sharp quantitative inequalities proved in Fusco et al. (2008) [7] and Fuglede (1989) [4] to reduce our problem to symmetric domains.

We establish comparison theorems involving exit time moments for Brownian motion and eigenvalues for the Dirichlet problem for domains in a complete Riemannian manifold. These theorems are closely related to isoperimetric properties of the ambient manifold.

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–Levy and Bakry–Ledoux isoperimetric inequalities, which state that whenever the curvature is strictly positively bounded from below, these model spaces are the n -sphere and Gauss space, corresponding to generalized dimension being n and ∞, respectively. In all other cases, which seem new even for the classical Riemannian-volume measure, it turns out that there is no single model space to compare to, and that a simultaneous comparison to a natural one parameter family of model spaces is required, nevertheless yielding a sharp result.

We introduce a new variational method for the study of stability in the isoperimetric inequality. The method is quite general as it relies on a penalization technique combined with the regularity theory for quasiminimizers of the perimeter. Two applications are presented. First we give a new proof of the sharp quantitative isoperimetric inequality in $R^n$. Second we positively answer to a conjecture by Hall concerning the best constant for the quantitative isoperimetric inequality in $R^2$ in the small asymmetry regime., Comment: 19 pages

We first estimate the containment measure of a convex domain to contain in another in a surface $$\mathbb{X}_\varepsilon$$ of constant curvature e. Then we obtain the analogue of the known Bonnesen isoperimetric inequality for convex domain in $$\mathbb{X}_\varepsilon$$ . Finally we strengthen the known Bonnesen isoperimetric inequality.

In this paper, we establish an isoperimetric inequality in a metric measure space via the Poisson equation. Let (X,d,μ) be a complete, pathwise connected metric space with locally Ahlfors Q-regular measure, where Q > 1, that supports a local L2-Poincare inequality. We show that, for the Poisson equation Δu = g, if the local L∞-norm of the gradient Du can be bounded by the Lorentz norm LQ,1 of g, then we obtain an isoperimetric inequality and a Sobolev inequality in (X,d,μ) with optimal exponents. By assuming a suitable curvature lower bound, we establish such optimal bounds on . © 2011 Wiley Periodicals, Inc.

We prove quantitative isoperimetric inequalities for images of the unit ball under homeomorphisms of exponentially integrable distortion. We show that the metric distortions of such domains can be controlled by their Fraenkel asymmetries. An application of the quantitative isoperimetric inequality proved by Hall and Fusco, Maggi, and Pratelli then shows that for these domains a version of Bonnesen’s inequality holds in all dimensions.

Let C be a closed convex set in a complete simply connected Riemannian manifold M with sectional curvature bounded above by a positive constant K. Assume that Σ is a compact minimal surface outside C such that Σ is orthogonal to ∂C along ∂Σ∩∂C and ∂Σ ∼ ∂C is radially connected from a point p ∈ ∂Σ∩∂C. We introduce a modified volume Mp(Σ) of Σ and obtain a sharp isoperimetric inequality where equality holds if and only if Σ is a geodesic half disk with constant Gaussian curvature K. We also prove higher dimensional isoperimetric inequalities for minimal submanifolds outside a closed convex set in a Riemannian manifold using the modified volume.

This paper is aimed at investigating some computational aspects of different isoperimetric problems on weighted trees. In this regard, we consider different connectivity parameters called minimum normalized cuts/isoperimetric numbers defined through taking the minimum of the maximum or the mean of the normalized outgoing flows from a set of subdomains of vertices, where these subdomains constitute a partition/subpartition. We show that the decision problem for the case of taking k-partitions and the maximum (called the max normalized cut problem NCPM), and the other two decision problems for the mean version (referred to as IPPm and NCPm) are NP-complete problems for weighted trees. On the other hand, we show that the decision problem for the case of taking k-subpartitions and the maximum (called the max isoperimetric problem IPPM) can be solved in linear time for any weighted tree and any k≥2. On the basis of this fact, we provide polynomial time O(k)-approximation algorithms for all different versions of the kth isoperimetric numbers considered.Moreover, for when the number of partitions/subpartitions, k, is a fixed constant, we prove, as an extension of a result of Mohar (1989) [20] for the case k=2 (usually referred to as the Cheeger constant), that the max and mean isoperimetric numbers of weighted trees, and their max minimum normalized cut can be computed in polynomial time. We also prove some hardness results for the case of simple unweighted graphs and trees.

In this paper, we deal with the reverse isoperimetric inequality for a closed and strictly convex curve in the Euclidean plane R 2 involving the following geometric functionals associated to the given convex curve: length, areas of the region respectively included by the curve and the locus of curvature centers, and the integral of the radius of curvature. In fact, a stronger and sharp version of the reverse isoperimetric inequality proved by Pan and Yang in (1) is established with a simple Fourier series proof. Furthermore, we investigate the stability property of such an inequality (almost equality implies that the curve is nearly circular).

We obtain an isoperimetric inequality which estimate the affine invariant p-surface area measure on convex bodies. We also establish the reverse version of L p -Petty projection inequality and an affine isoperimetric inequality of Γ − p K.

We solve the isoperimetric problem, the least-perimeter way to enclose a given area, on various Euclidean, spherical, and hyperbolic surfaces, sometimes with cusps or free boundary. On hyperbolic genus-two surfaces, Adams and Morgan characterized the four possible types of isoperimetric regions. We prove that all four types actually occur and that on every hyperbolic genus-two surface, one of the isoperimetric regions must be an annulus. In a planar annulus bounded by two circles, we show that the leastperimeter way to enclose a given area is an arc against the outer boundary or a pair of spokes. We generalize this result to spherical and hyperbolic surfaces bounded by circles, horocycles, and other constant-curvature curves. In one case the solution alternates back and forth between two types, a phenomenon we have yet to see in the literature. We also examine non-orientable surfaces such as spherical Mobius bands and hyperbolic twisted chimney spaces.

In this paper, we provide various Sobolev-type inequalities for smooth nonnegative functions with compact support on a submanifold with variable mean curvature in a Riemannian manifold whose sectional curvature is bounded above by a constant. We further obtain the corresponding linear isoperimetric inequalities involving mean curvature. We also provide various first Dirichlet eigenvalue estimates for submanifolds with bounded mean curvature.

We find approximate solutions (chord–arc curves) for the system of equations of geodesics in Ω∩ℍ for every Denjoy domain Ω, with respect to both the Poincaré and the quasi-hyperbolic metrics. We also prove that these chord–arc curves are uniformly close to the geodesics. As an application of these results, we obtain good estimates for the lengths of simple closed geodesics in any Denjoy domain, and we improve the characterization in a 1999 work by Alvarez et al. on Denjoy domains satisfying the linear isoperimetric inequality.

The isoperimetric problem with respect to the product-type density e − | x | 2 2 d x d y on the Euclidean space R h × R k is studied. In particular, existence, symmetry and regularity of minimizers is proved. In the special case k = 1 , also the shape of all the minimizers is derived. Finally, a conjecture about the minimality of large cylinders in the case k > 1 is formulated.

In this paper, we obtain the isoperimetric inequality on a conformally flat manifold with finite total Q-curvature. This is a higher dimensional analog of Li and Tam’s result [“Complete surfaces with finite total curvature.” Journal of Differential Geometry 33 (1991): 139–68] on surfaces with finite total Gaussian curvature. The main step in the proof is based on the construction of a quasiconformal map the Jacobian of which is suitably bounded.