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Functional inequalities related to Mahler's conjecture

Abstract : Extending a result of Meyer and Reisner (Monatsh Math 125: 219-227, 1998), we prove that if g : R -> R(+) is a function which is concave on its support, then for every m > 0 and every z is an element of R such that g(z) > 0, one has integral(R) g(x)(m)dx integral(R) (g*(z)(y))(m)dy >= (m + 2)(m+2)/(m + 1)(m+3), where for y is an element of R, g*(z)(y) = inf(x) (1-(x-z)y)+/g(x). It is shown how this inequality is related to a special case of Mahler's conjecture (or inverse Santalo inequality) for convex bodies. The same ideas are applied to give a new (and simple) proof of the exact estimate of the functional inverse Santalo inequality in dimension 1 given in Fradelizi and Meyer (Adv Math 218: 1430-1452, 2008). Namely, if phi : R -> R boolean OR {+infinity} is a convex function such 0 < integral e(-phi) < +infinity then, for every z is an element of R such that phi(z) < +infinity, one has integral(R) e(-phi) integral(R) e(-Lz phi) >= e, where L(z)phi is the Legendre transform of phi with respect to z.
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Submitted on : Tuesday, May 1, 2012 - 7:42:00 PM
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Matthieu Fradelizi, Mathieu Meyer. Functional inequalities related to Mahler's conjecture. Monatshefte für Mathematik, Springer Verlag, 2010, 159 (1-2), pp.13--25. ⟨10.1007/s00605-008-0064-0⟩. ⟨hal-00693038⟩