We give a functional version of the affine isoperimetric inequality for log-concave functions which may be interpreted as an inverse form of a logarithmic Sobolev inequality for entropy. A linearization of this inequality gives an inverse inequality to the Poincaré inequality for the Gaussian measure.
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* Corresponding author at: Department of Mathematics, Case Western Reserve University, Cleveland, OH 44106, USA. E-mail addresses: email@example.com (S. Artstein-Avidan), firstname.lastname@example.org (B. Klartag), email@example.com (C. Schütt), firstname.lastname@example.org (E. Werner). 1 Partially supported by BSF grant No. 2006079 and by ISF grant No. 865/07. 2 Partially supported by an ISF grant and an IRG grant. 3 Partially supported by NSF grant and BSF grant No. 2006079.
- Affine isoperimetric inequality
- Logarithmic Sobolev inequality