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Increasing functions and inverse Santalo inequality for unconditional functions

Abstract : Let phi : R(n) -> R boolean OR {+infinity} be a convex function and L phi be its Legendre tranform. It is proved that if phi is invariant by changes of signs, then integral e(-phi) integral e(-L phi) >= 4(n). This is a functional version of the inverse Santalo inequality for unconditional convex bodies due to J. Saint Raymond. The proof involves a general result on increasing functions on R(n) x 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 integral e-(phi) integral e(-L phi) >= c(n). This generalizes a result of B. Klartag and V. Milman.
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https://hal-upec-upem.archives-ouvertes.fr/hal-00693067
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Submitted on : Tuesday, May 1, 2012 - 7:54:05 PM
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Matthieu Fradelizi, Mathieu Meyer. Increasing functions and inverse Santalo inequality for unconditional functions. Positivity, Springer Verlag, 2008, 12 (3), pp.407--420. ⟨10.1007/s11117-007-2145-z⟩. ⟨hal-00693067⟩

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