We continue our investigation on the transportation-information inequalities WpI for a symmetric markov process, introduced and studied in [14]. We prove that WpI implies the usual transportation inequalities WpH, then the corresponding concentration inequalities for the invariant measure μ. We give also a direct proof that the spectral gap in the space of Lipschitz functions for a diffusion pr...

The Brunn-Minkowski inequality gives a lower bound on the Lebesgue measure of a sumset in terms of the measures of the individual sets. This inequality plays a crucial role in the theory of convex bodies and has many interactions with isoperimetry and functional analysis. Stability of optimizers of this inequality in one dimension is a consequence of classical results in additive combinatorics....

in this paper, we study the chebyshev centres of bounded subsets of normed spaces and obtain a norm inequality for relative centres. in particular, we prove that if t is a remotal subset of an inner product space h, and f is a star-shaped set at a relative chebyshev centre c of t with respect to f, then llx - qt (x)1i2 2 ilx-cll2 + ilc-qt (c) 112 x e f, where qt : f + t is any choice function s...

This paper presents and characterizes a two-parameter class of inequality measures that contains the generalized entropy measures, the variance of logarithms, the path independent measures of Foster and Shneyerov (1999) and several new classes of measures. The key axiom is a generalized form of additive decomposability which defines the within-group and between-group inequality terms using a ge...

We present some recent stability results concerning the isoperimetric inequality and other related geometric and functional inequalities. The main techniques and approaches to this field are discussed.

* Correspondence: [email protected]. zj.cn School of Mathematical Sciences, Xiamen University, Xiamen, 361005, PR China Abstract In this article, a new nonlinear impulsive delay differential inequality is established, which can be applied in the dynamical analysis of nonlinear systems to improve many extant results. Using the inequality, we obtain some sufficient conditions to guarantee the exponen...

In this paper we introduce and study a weakened form of logarithmic Sobolev inequalities in connection with various others functional inequalities (weak Poincaré inequalities, general Beckner inequalities...). We also discuss the quantitative behaviour of relative entropy along a symmetric diffusion semi-group. In particular, we exhibit an example where Poincaré inequality can not be used for d...

In this paper we prove a sharp version of the Moser-Trudinger inequality for the Euler-Lagrange functional of a singular Toda system, motivated by the study of models in Chern-Simons theory. Our result extends those in [14] and [37] for the scalar case, as well as that in [23] for the regular Toda system. We expect this inequality to be a basic tool to attack variationally the existence problem...