Increasing functions and inverse Santalo inequality for unconditional functions

Matthieu Fradelizi (1) , Mathieu Meyer (1)

1 LAMA - Laboratoire d'Analyse et de Mathématiques Appliquées

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.

Domaines: Physique mathématique [math-ph]

Matthieu Fradelizi, Mathieu Meyer. Increasing functions and inverse Santalo inequality for unconditional functions. Positivity, 2008, 12 (3), pp.407--420. ⟨10.1007/s11117-007-2145-z⟩. ⟨hal-00693067⟩