Isoperimetric Type Inequalities Research Articles

TWO COMPLEX COMBINATIONS AND COMPLEX INTERSECTION BODIES

This paper devotes to establish complex dual Brunn-Minkowski theory. At first, we introduce the concepts of complex radial combination and complex radial-Blaschke combination, and obtain the relations between those two combinations and dual mixed volumes. Then, we extend the properties of real intersection body to the complex case. Finally, we prove some complex geometric inequalities about complex intersection bodies and complex mixed intersection bodies, such as dual Brunn-Minkowski type, dual Aleksandrov-Fenchel type and dual Minkowski type inequality. Moreover, as applications, we get some corollaries including an isoperimetric type inequality and a uniqueness theorem.

From symplectic measurements to the Mahler conjecture

In this note we link symplectic and convex geometry by relating two seemingly different open conjectures: a symplectic isoperimetric-type inequality for convex domains and Mahler’s conjecture on the volume product of centrally symmetric convex bodies. More precisely, we show that if for convex bodies of fixed volume in the classical phase space the Hofer–Zehnder capacity is maximized by the Euclidean ball, then a hypercube is a minimizer for the volume product among centrally symmetric convex bodies.

Volume differences of mixed complex projection bodies

Recently, Abardia and Bernig introduced the notion of mixed complex projection bodies and established a number of important geometric inequalities for them. In the present paper we prove several new isoperimetric type inequalities for volume differences of mixed complex projection bodies.

Random chord distributions and containment functions

Formulas for the probability distribution function of the length of random chord in a convex body intersected by a random line are established by using the containment function and the chord projection function of the convex body. Explicit expressions of random chord distributions for several special convex bodies are given. An isoperimetric-type inequality for the moments of random chord is proved. Similar results for random distance distribution of two random points in a convex body are also obtained.

The isoperimetric type and Fejer-Riesz type inequalities for pluriharmonic mappings

We investigate the isoperimetric type and Fejer-Riesz type inequalities for pluriharmonic mappings. Firstly, we improve the corresponding result of Kalaj and Mestrovic, and then we generalize it to pluriharmonic mappings. Secondly, we extend the corresponding result of Pavlovic and Dostanic to pluriharmonic mappings. Finally, we establish the Fejer-Riesz type inequality for pluriharmonic mappings.

On radial Blaschke-Minkowski homomorphisms

In this article we first establish a dual isoperimetric type inequality for mixed radial Blaschke-Minkowski homomorphisms of different orders. Second, we prove a dual Brunn-Minkowski-type and a dual Minkowski-type inequality for mixed radial Blaschke-Minkowski homomorphisms.

Stability of isoperimetric type inequalities for some Monge–Ampère functionals

We investigate the stability of some inequalities of isoperimetric type related to Monge–Ampere functionals. In particular, firstly we prove the stability of a reverse Faber–Krahn inequality for the Monge–Ampere eigenvalue and its generalization. Then we give a stability result for the Brunn–Minkowski inequality and for a consequent Urysohn’s type inequality for the so-called \(n\)-torsional rigidity, a natural extension of the usual torsional rigidity.

A probabilistic take on isoperimetric-type inequalities

We extend a theorem of Groemer on the expected volume of a random polytope in a convex body. The extension involves various ways of generating random convex sets. We also treat the case of absolutely continuous probability measures rather than convex bodies. As an application, we obtain a new proof of a recent result due to Lutwak, Yang and Zhang on the volume of Orlicz-centroid bodies.

Isoperimetric-type inequalities on constant curvature manifolds

By exploiting optimal transport theory on Riemannian manifolds and adapting Gromov’s proof of the isoperimetric inequality in the Euclidean space, we prove an isoperimetric-type inequality on simply connected constant curvature manifolds.

Un nuovo approccio alle disuguaglianze isoperimetriche quantitative

We introduce a new variational method for studying geometric and functional inequalities with quantitative terms. In the context of isoperimetric-type inequalities, this method (called Selection Principle) is based on a penalization technique combined with the regularity theory of quasiminimizers of the perimeter functional. In this seminar we present the method and describe two remarkable applications. The rst one is a new proof of the sharp quantitative isoperimetric inequality in Rn. The second one is the proof of a conjecture posed by Hall about the optimal constant in the quantitative isoperimetric inequality in R2, in the small asymmetry regime.

A note on the isoperimetric inequality and its stability

In this paper, we deals with isoperimetric-type inequalities for closed convex curves in the Euclidean plane R 2 . We derive a family of parametric inequalities involving the fol- lowing geometric functionals associated to a given convex curve with a simple Fourier series proof: length, area of the region included by the curve, area of the domain enclosed by the lo- cus of curvature centers and integral of the radius of curvature. By using our isoperimetric-type inequalities, we also obtain some new geometric Bonnesen-type inequalities. Furthermore we investigate stability properties of such inequalities (near equality implies curve nearly circular).

Isoperimetric type inequalities for harmonic functions

For 0 < p < + ∞ let h p be the harmonic Hardy space and let b p be the harmonic Bergman space of harmonic functions on the open unit disk U . Given 1 ⩽ p < + ∞ , denote by ‖ ⋅ ‖ b p and ‖ ⋅ ‖ h p the norms in the spaces b p and h p , respectively. In this paper, we establish the harmonic h p -analogue of the known isoperimetric type inequality ‖ f ‖ b 2 p ⩽ ‖ f ‖ h p , where f is an arbitrary holomorphic function in the classical Hardy space H p . We prove that for arbitrary p > 1 , every function f ∈ h p satisfies the inequality ‖ f ‖ b 2 p ⩽ a p ‖ f ‖ h p , where a p > 1 is a suitable constant depending only on p. Furthermore, by using the Carleman inequality in the form ‖ f ‖ b 4 ⩽ ‖ f ‖ h 2 with f ∈ H 2 , we prove the following refinement of the above inequality for p = 2 ‖ f ‖ b 4 ⩽ 1.5 + 2 4 ‖ f ‖ h 2 .

On Quasiconformal Harmonic Surfaces with Rectifiable Boundary

It is proved that every quasiconfomal harmonic mapping of the unit disk onto a surface with rectifiable boundary has absolutely continuous extension to the boundary as well as its inverse mapping has this property. In addition it is proved an isoperimetric type inequality for the class of these surfaces. These results extend some classical results for conformal mappings, minimal surfaces and surfaces with constant mean curvature treated by Kellogg, Courant, Nitsche, Tsuji, F. Riesz and M. Riesz, etc.

Bounds for inclusion measures of convex bodies

By using the method of mixed volumes, we give sharp bounds for inclusion measures of convex bodies in n-dimensional Euclidean space. In the special cases where the random convex body is the unit ball or when n = 3 , neater and simpler bounds are obtained. All the associated inequalities proved are new isoperimetric-type inequalities.

Minimal webs in Riemannian manifolds

For a given combinatorial graph G a geometrization (G, g) of the graph is obtained by considering each edge of the graph as a 1-dimensional manifold with an associated metric g. In this paper we are concerned with minimal isometric immersions of geometrized graphs (G, g) into Riemannian manifolds (N n ,h ). Such immersions we call minimal webs. They admit a natural 'geometric' extension of the intrinsic combinatorial discrete Laplacian. The geometric Laplacian on minimal webs enjoys standard properties such as the maximum principle and the divergence theorems, which are of instrumental importance for the appli- cations. We apply these properties to show that minimal webs in ambient Riemannian spaces share several analytic and geometric properties with their smooth (minimal submanifold) counterparts in such spaces. In particular we use appropriate versions of the divergence theo- rems together with the comparison techniques for distance functions in Riemannian geometry and obtain bounds for the first Dirichlet eigenvalues, the exit times and the capacities as well as isoperimetric type inequalities for so-called extrinsic R-webs of minimal webs in ambient Riemannian manifolds with bounded curvature.

Chord power integrals and radial mean bodies

In this paper, we obtain a formula relating the chord power integrals of a convex body K and the dual quermassintegrals of its radial pth mean body R p K . With this, a relation among the chord power integrals of a convex body K under dilation transformations is found. As an interesting application, some geometric inequalities between the dual quermassintegrals of R p K and the volume of K, which are equivalent to the isoperimetric-type inequalities of chord power integrals, are also established.

Isoperimetric-type inequalities for iterated Brownian motion in [formula omitted

We extend generalized isoperimetric-type inequalities to iterated Brownian motion over several domains in R n . These kinds of inequalities imply in particular that for domains of finite volume, the exit distribution and moments of the first exit time for iterated Brownian motion are maximized with the ball D * centered at the origin, which has the same volume as D.

Asymptotic Shapes of large Cells in Random Tessellations

We establish a close relationship between isoperimetric inequalities for convex bodies and asymptotic shapes of large random polytopes, which arise as cells in certain random mosaics in d-dimensional Euclidean space. These mosaics are generated by Poisson hyperplane processes satisfying a few natural assumptions (not necessarily stationarity or isotropy). The size of large cells is measured by a class of general functionals. The main result implies that the asymptotic shapes of large cells are completely determined by the extremal bodies of an inequality of isoperimetric type, which connects the size functional and the expected number of hyperplanes of the generating process hitting a given convex body. We obtain exponential estimates for the conditional probability of large deviations of zero cells from asymptotic or limit shapes, under the condition that the cells have large size.

On geometric properties of directed vertex-symmetric graphs

Let Γ be a directed locally finite vertex-symmetric graph such that the underlying graph Γ ¯ is connected. We prove that, in the case Γ is infinite, there exists a positive integer k , depending only on min { deg + ( Γ ) , deg − ( Γ ) } , such that for any positive integer n there exists a directed path of Γ of length not greater than k n 2 whose initial and terminal vertices are at distance n in the graph Γ ¯ . We also prove an analogous result in the case Γ is finite and n < diam ( Γ ¯ ) 2 + 1 . The results are deduced from certain isoperimetric type inequalities for cones in Γ .

Isoperimetric type inequalities for differential forms on manifolds

Let X be a smooth oriented Riemannian n-manifold without boundary and (Φ,Ψ) ∈ L p (Λ X ) × L r (Λ n-X ), 1/p+ 1/r = 1 + 1/n, be a pair of closed differential forms. We prove an isoperimetric type inequality for such differential forms under suitable assumptions. As an application we derive Holder continuity for solutions of Hodge systems.