Abstract

We discuss some refinements of the classical Prékopa-Leindler inequality, which consist in the addition of an extra-term depending on a distance modulo translations. Our results hold true on suitable classes of functions of n variables. They are based upon two different kinds of 1-dimensional refinements: the former is the one obtained by K. M. Ball and K. Böröczky ["Stability of the Prékopa-Leindler inequality", Mathematika 56 (2010) 339-356] and involves an L1L^1-type distance on log-concave functions, the latter is new and involves the transport map onto the Lebesgue measure. Starting from each of these 1-dimensional refinements, we obtain an n-dimensional counterpart by exploiting a generalized version of the Cramér-Wold Theorem.

Author information

Contact details are reproduced from the original publication and may be historical.

Dorin Bucur

Lab. de Mathématiques, Université de Savoie, Campus Scientifique, 73376 Le-Bourget-Du-Lac, France

dorin.bucur@univ-savoie.fr

Ilaria Fragalà

Dip. di Matematica, Politecnico di Milano, Piazza Leonardo da Vinci 32, 20133 Milano, Italy

ilaria.fragala@polimi.it

Keywords

  • Functional inequalities
  • Cramér-Wold Theorem
  • log-concave functions
  • mass transportation

Mathematics Subject Classification

52A40, 26D10, 39B62

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Suggested citation

D. Bucur, I. Fragalà. “Lower Bounds for the Prékopa-Leindler Deficit by Some Distances Modulo Translations.” Journal of Convex Analysis 21 (2014), No. 1, 289–305. https://doi.org/10.68381/jca21016

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Published by Heldermann Verlag, 2014. Rights now held by Banach Press LLC.