Some functional forms of Blaschke–Santaló inequality

By Matthieu Fradelizi, Mathieu Meyer. 2006

131 citations recorded by the source corpus.

We establish new functional versions of the Blaschke–Santaló inequality on the volume product of a convex body which generalize to the non-symmetric setting an inequality of Ball [Isometric problems in ℓp and sections of convex sets. PhD Dissertation, Cambridge, 1986] and we give a simple proof of the case of equality. As a corollary, we get some inequalities for log-concave functions and Legendre transforms which extend the recent result of Artstein et al. [Mathematika 51:33–48, 2004], with its equality case.

Subfields: Applied Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/e8b6b9e83c417b98d4afe0ca75ed01a214ec5ba4

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In Applied Mathematics

In Geometry and Topology

Citations in the corpus

A characterization of the concept of duality (2007, background): “This was prompted by the joint work in [4], where the definition for functional duality from Example 3 above was suggested, which seemed “right” in the sense that the inequalities it anticipated turned out to be either known, or provable.”
Functional affine-isoperimetry and an inverse logarithmic Sobolev inequality (2011, background): “1 Introduction There is a general approach to extend invariants of convex bodies to the corresponding invariants of functions [1, 7, 10, 17].”
Partitions and functional Santaló inequalities (2009, methodology): “The following lemma is a useful (see [4, 2]) logarithmic version of Prékopa-Leindler.”
Some functional inverse Santalo inequalities (2008, background): “Then the functional Blaschke–Santaló inequality, proved for even functions in [2], and given in full generality in [1] (see also [7] for generalizations), states that P(f ) " P ( e− ‖·‖(2)2 2 ) , where ‖ · ‖2 stands here for the Euclidean norm in R.”
A direct proof of the functional Santalo inequality (2010, result): “In the special case where the function f is even, this result follows from an earlier inequality of Keith Ball [2]; and in [4], Fradelizi and Meyer prove something more general (see also [5]).”
Increasing functions and inverse Santaló inequality for unconditional functions (2008, background): “Ball [2] and in full generality by Artstein-Klartag-Milman [1] (see also Fradelizi-Meyer [8]).”
Divergence for s-concave and log concave functions (2013, methodology): “The corresponding inequalities for log concave functions were proved by Ball [7] and Artstein, Klartag and Milman [5] (see also [17, 31]).”
Rogers–Shephard inequality for log-concave functions (2014, background): “This has generated an increasing interest to extend several important parameters of convex bodies to functional parameters [5,6,10,12,19,22,24] in the class of log-concave functions.”
Remarks on the conjectured log-Brunn–Minkowski inequality (2013, unknown): “(9) Next, in the same spirit as in [15], Proposition 1, use the change of variables zi 7→ e zi , i = 1, .”