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Positivity

, Volume 12, Issue 3, pp 407–420 | Cite as

Increasing functions and inverse Santaló inequality for unconditional functions

  • Matthieu FradeliziEmail author
  • Mathieu Meyer
Article

Abstract

Let \(\phi: {\mathbb{R}}^n\to {\mathbb{R}}\cup\{+\infty\}\) be a convex function and \(\mathcal{L}\phi\) be its Legendre tranform. It is proved that if \(\phi\) is invariant by changes of signs, then \(\int e^{-\phi}\int e^{-\mathcal{L}\phi} \ge 4^n\). This is a functional version of the inverse Santaló inequality for unconditional convex bodies due to J. Saint Raymond. The proof involves a general result on increasing functions on \(\mathbb{R}^{n} \times \mathbb{R}^n\) together with a functional form of Lozanovskii’s lemma. In the last section, we prove that for some c > 0, one has always \(\int e^{-\phi}\int e^{-\mathcal{L}\phi} \ge c^n\). This generalizes a result of B. Klartag and V. Milman.

Mathematics Subject Classification (2000)

Primary: 52A40 26B25 39B62 

Keywords

Increasing functions Legendre transform inverse-Santaló inequality 

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Copyright information

© Springer Science + Business Media B.V. 2008

Authors and Affiliations

  1. 1.Université de Marne la Vallée, Laboratoire d’Analyse et de Mathématiques Appliquées (UMR 8050)Marne la Vallée Cedex 2France

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