Modified Log-sobolev Inequalities and Isoperimetry

We find sufficient conditions for a probability measure μ to satisfy an inequality of the type ∫ Rd fF ( f ∫ Rd f 2 dμ ) dμ ≤ C ∫ Rd f2c∗ ( |∇f | |f | ) dμ + B ∫ Rd f dμ, where F is concave and c (a cost function) is convex. We show that under broad assumptions on c and F the above inequality holds if for some δ > 0 and ε > 0 one has ∫ ε 0 Φ ( δc [ tF ( 1t ) Iμ(t) ]) dt <∞, where Iμ is the isoperimetric function of μ and Φ = (yF (y)− y)∗. In a partial case Iμ(t) ≥ ktφ1− 1 α (1/t), where φ is a concave function growing not faster than log, k > 0, 1 < α ≤ 2 and t ≤ 1/2, we establish a family of tight inequalities interpolating between the F -Sobolev and modified inequalities of log-Sobolev type. A basic example is given by convex measures satisfying certain integrability assumptions.