# Increasing functions and inverse Santaló inequality for unconditional functions

Increasing functions and inverse Santaló inequality for unconditional functions 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. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Positivity Springer Journals

# Increasing functions and inverse Santaló inequality for unconditional functions

, Volume 12 (3) – Mar 1, 2008
14 pages

/lp/springer_journal/increasing-functions-and-inverse-santal-inequality-for-unconditional-OhZMae1n0g
Publisher
Springer Journals
Subject
Mathematics; Econometrics; Calculus of Variations and Optimal Control; Optimization; Potential Theory; Operator Theory; Fourier Analysis
ISSN
1385-1292
eISSN
1572-9281
D.O.I.
10.1007/s11117-007-2145-z
Publisher site
See Article on Publisher Site

### 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.

### Journal

PositivitySpringer Journals

Published: Mar 1, 2008

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