Modified log-Sobolev inequalities and isoperimetry

  • Alexander V. Kolesnikov

    Moscow State University, Russian Federation


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. 179–208

DOI 10.4171/RLM/489