A variant of Jensen-Steffensen’s inequality is considered for convex and for superquadratic functions. Consequently, inequalities for power means involving not only positive weights have been established.

We prove a vertex-isoperimetric inequality for [n], the set of all r-element subsets of {1, 2, . . . , n}, where x, y ∈ [n] are adjacent if |x∆y| = 2. Namely, if A ⊂ [n] with |A| = α ( n r ) , then its vertex-boundary b(A) satisfies |b(A)| ≥ c √ n r(n− r)α(1− α) ( n r )

For a smooth curve γ, we define its elastic energy as E(γ) = 12 ∫ γ k(s)ds where k(s) is the curvature. The main purpose of the paper is to prove that among all smooth, simply connected, bounded open sets of prescribed area in R, the disc has the boundary with the least elastic energy. In other words, for any bounded simply connected domain Ω, the following isoperimetric inequality holds: E(∂Ω)...

An uncertain random variable is a measurable function from a probability space to the set of uncertain variables. In this paper, subadditivity property of chance measure and some inequalities on expected value for uncertain random variables including Jensen’s Inequality, Liapunov Inequality and cp Inequality are proved. Based on these inequalities, linear property and other properties of conver...

To verify whether a transferable utility game is exact, one has to check a linear inequality for each exact balanced collection of coalitions. This paper studies the structure and properties of the class of exact balanced collections. Comparing the definition of exact balanced collections with the definition of balanced collections, the weight vector of a balanced collection must be positive wh...

In this paper we study the quantitative isoperimetric inequality in the plane. We prove the existence of a set Ω, different from a ball, which minimizes the ratio δ(Ω)/λ(Ω), where δ is the isoperimetric deficit and λ the Fraenkel asymmetry, giving a new proof of the quantitative isoperimetric inequality. Some new properties of the optimal set are also shown.

A new refinement of Jensen’s celebrated inequality for functions defined on convex sets in linear spaces is given. Applications for norms, mean f -deviation and f -divergences are provided as well.

In this paper, the concept of the classical f -divergence for a pair of measures is extended to the mixed f -divergence for multiple pairs of measures. The mixed f -divergence provides a way to measure the difference between multiple pairs of (probability) measures. Properties for the mixed f -divergence are established, such as permutation invariance and symmetry in distributions. An Alexandro...

We prove that Friedlander’s generalized isomorphism conjecture on the cohomology of algebraic groups, and hence Milnor’s conjecture on the cohomology of the complex algebraic Lie group G(C) made discrete, are equivalent to the existence of an isoperimetric inequality in the homological bar complex of G(F ), where F is the algebraic closure of a finite field.

If F is a finitely generated free group and φ is an automorphism of F then F ⋊φZ satisfies a quadratic isoperimetric inequality.