The Isoperimetric Inequality in Steady Ricci Solitons

The author proves that the isoperimetric inequality on the graphic curves over circle or hyperplanes over S^n-1is satisfied in the cigar steady soliton and in the Bryant steady soliton. Since both of them are Riemannian manifolds with warped product metric,the author utilize the result of Guan-Li-Wang to get his conclusion. For the sake of the soliton structure, the author believes that the geometric restrictions for manifolds in which the isoperimetric inequality holds are naturally s...     

两平面凸域的对称混合等周不等式

设K_k(k=i,j)为欧氏平面R~2中面积为A_k,周长为P_k的域,它们的对称混合等周亏格(symmetric mixed isoperimetric deficit)为σ(K_i,K_j)=P_i~2P_j~2-16π~2A_iA_j.根据周家足,任德麟(2010)和Zhou,Yue(2009)中的思想,用积分几何方法,得到了两平面凸域的Bonnesen型对称混合不等式及对称混合等周不等式,给出了两域的对称混合等周亏格的一个上界估计.还得到了两平面凸域的离散Bonnesen型对称混合不等式及两凸域的对称混合等周亏格的一个上界估计,并应用这些对称混合(等周)不等式估计第二类完全椭圆积分.     

Isoperimetric Inequalities on Compact Manifolds

We prove an isoperimetric inequality on compact Riemannian manifolds corresponding to the limit case of a scale of optimal Sobolev inequalities.     

The Isoperimetric Inequality on Manifolds of Conformally Hyperbolic Type

We prove that the maximal isoperimetric function on a Riemannian manifold of conformally hyperbolic type can be reduced to the linear canonical form P(x) = x by a conformal change of the Riemannian metric. In other words, the isoperimetric inequality , relating the volume V(D) of a domain D to the area of its boundary, can be reduced to the form , known for the Lobachevskii hyperbolic space.     

The Gaussian Isoperimetric Inequality and Transportation

Any probability measure on ^d which satisfies the Gaussian or exponential isoperimetric inequality fulfils a transportation inequality for a suitable cost function. Suppose that W (x) dx satisfies the Gaussian isoperimetric inequality: then a probability density function f with respect to W (x) dx has finite entropy, provided that t . This strengthens the quadratic logarithmic Sobolev inequality of Gross (Amr. J. Math 97 (1975) 1061). Let (dx) = e ^–(x) dx be a probability measure on ^d, where is uniformly convex. Talagrand's technique extends to monotone rearrangements in several dimensions (Talagrand, Geometric Funct. Anal. 6 (1996) 587), yielding a direct proof that satisfies a quadratic transportation inequality. The class of probability measures that satisfy a quadratic transportation inequality is stable under multiplication by logarithmically bounded Lipschitz densities.     

Isoperimetric inequality for radial probability measures on Euclidean spaces

We generalize the Poincaré 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.     

Isoperimetric type inequalities for harmonic functions

For 0<p<+∞ let hp be the harmonic Hardy space and let bp be the harmonic Bergman space of harmonic functions on the open unit disk U. Given 1?p<+∞, denote by ‖⋅bp_‖ and ‖⋅hp_‖ the norms in the spaces bp and hp, respectively. In this paper, we establish the harmonic hp-analogue of the known isoperimetric type inequality ‖fb2p_‖?‖fhp_‖, where f is an arbitrary holomorphic function in the classical Hardy space Hp. We prove that for arbitrary p>1, every function f∈hp satisfies the inequality

‖fb2p_‖?ap‖fhp_‖,     

体积差的等周不等式

首次提出并建立了凸体的体积差函数的等周不等式,它是经典等周不等式的推广.作为应用,对星体建立了体积差函数的对偶等周不等式和广义对偶等周不等式.     

Projection Body and Isoperimetric Inequalities for s-Concave Functions*

For a positive integer s, the projection body of an s-concave function f : Rⁿ →[0, +∞), a convex body in the (n + s)-dimensional Euclidean space R^n+s, is introduced.Associated inequalities for s-concave functions, such as, the functional isoperimetric inequality, the functional Petty projection inequality and the functional Loomis-Whitney inequality are obtained.     

半线性椭圆方程正解的等周不等式

证明了一类半线性椭圆方程正解满足等周不等式,并得到了此解的最佳上界估计.     

Mass-Capacity Inequalities for Conformally Flat Manifolds with Boundary

In this paper we prove a mass-capacity inequality and a volumetric Penrose inequality for conformally flat manifolds, in arbitrary dimensions. As a by-product of the proofs, capacity and Aleksandrov-Fenchel inequalities for mean-convex Euclidean domains are obtained. For each inequality, the case of equality is characterized.     

Beckner Inequality on Finite- and Infinite-Dimensional Manifolds

By using the dimension-free Harnack inequality, the coupling method, and Bakry-Emery's argument, some explicit lower bounds are presented for the constant of the Beckner type inequality on compact manifolds. As applications, the Beckner inequality and the transportation cost inequality are established for a class of continuous spin systems. In particular, some results in [1, 2] are generalized.     

Variational Inequality with Fuzzy Convex Cone

In this study, we discuss one type of variational inequality problem with a fuzzy convex cone , denoted by VI( , f). Different classes of fuzzy convex cones which are considered in different context of the problems will be discussed. According to the existence theorem, an approach derived from the concepts of multiple objective mathematical programming problems for solving the VI( , f) is proposed. An algorithm is developed to find its fuzzy optimal solution set with complexity analysis.     

Convex Hypersurfaces with Prescribed Curvature and Boundary in Riemannian Manifolds

In this article a priori estimates at the boundary for the second fundamental form of n-dimensional convex hypersurfaces M with prescribed curvature quotient Sn (M)/Sl (M) in Riemannian manifolds are derived. A consequence of these estimates and other known results is an existence theorem for such hypersurfaces, which is a generalization of a recent result of Ivochkina and Tomi to the Riemannian case.     

An Isoperimetric Inequality for Hyperplane Sections of a Convex Body

We obtain an inequality which estimates the product of the volumes of the two parts into which a convex body is divided by a hyperplane through its interior in terms of the volume of the hyperplane section. The basic tool in the proof is a Localization lemma for log-concave functions due to Kannan, Lovász and Simonovits.     

Isoperimetric curves on hyperbolic surfaces

Least-perimeter enclosures of prescribed area on hyperbolic surfaces are characterized.

     

Isoperimetric estimates in products

In a product M ₁ × M ₂ of Riemannian manifolds, the least perimeter required to enclose given volume among general regions is at least 1/√ 2 times that among regions of product form, assuming that the isoperimetric profiles of M ₁ and M ₂ are concave. This result sharpens earlier work of Grigor'yan, generalizes results of Bollobás and Leader and of Barthe, and yields a lower bound on the Cheeger isoperimetric constant of a product.     

Infinite Dimensional Isoperimetric Inequalities in Product Spaces with the Supremum Distance

We study the isoperimetric problem for product probability measures with respect to the uniform enlargement. We construct several examples of measures for which the isoperimetric function of coincides with the one of the infinite product . This completes earlier works by Bobkov and Houdré.     

The Logarithmic Sobolev Inequality for a Submanifold in Manifolds with Nonnegative Sectional Curvature

The authors prove a sharp logarithmic Sobolev inequality which holds for compact submanifolds without boundary in Riemannian manifolds with nonnegative sectional curvature of arbitrary dimension and codimension. Like the Michael-Simon Sobolev inequality, this inequality includes a term involving the mean curvature. This extends a recent result of Brendle with Euclidean setting.     

The Hadamard Inequality For Convex Function Via Fractional Integrals

In this paper, we establish several inequalities for some differantiable mappings that are connected with the Riemann-Liouville fractional integrals. The analysis used in the proofs is fairly elementary.