We find sufficient conditions for a probability measure to satisfy an inequality of the type
where is concave and (a~cost function) is convex. We show that under broad assumptions on and the above inequality holds if for some and one has
where is the isoperimetric function of and . In a partial case
where is a concave function growing not faster than , , and , we establish a family of tight inequalities interpolating between the -Sobolev and modified inequalities of log-Sobolev type. A basic example is given by convex measures satisfying certain integrability assumptions.
Cite this article
Alexander V. Kolesnikov, Modified log-Sobolev inequalities and isoperimetry. Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. 18 (2007), no. 2, pp. 179–208DOI 10.4171/RLM/489