We consider coercive and isoperimetric inequalities for probability measures with super-Gaussian tails. First part of the work is devoted to analysis of modified forms of logarithmic Sobolev inequalities, which we call LSq, on abstract (metric) spaces with respect to Lq-norm of the modulus of gradient. We give an explicit characterization of probability distributions satisfying these inequalities on the real line. It is also shown that LSg are satisfied for a large class of non-trivial infinite dimensional measures (such as Gibbs measures). In the second part, Sobolev-type inequalities are studied in the class of uniform distributions on convex bodies in finite dimensional Euclidean spaces. In particular, we refine a result of Kannan, Lovász and Simonovits on the Cheeger isoperimetric constants.
- Coercive and isoperimetric inequalities
- Concentration of measure
- Gibbs measures
- Infinite dimensional spaces
- Markov semigroups