Mixed integrals and related inequalities

By V. Milman, Liran Rotem. 2012

58 citations recorded by the source corpus.

Abstract In this paper we define an addition operation on the class of quasi-concave functions. While the new operation is similar to the well-known sup-convolution, it has the property that it polarizes the Lebesgue integral. This allows us to define mixed integrals, which are the functional analogs of the classic mixed volumes. We extend various classic inequalities, such as the Brunn–Minkowski and the Alexandrov–Fenchel inequalities, to the functional setting. For general quasi-concave functions, this is done by restating those results in the language of rearrangement inequalities. Restricting ourselves to log-concave functions, we prove generalizations of the Alexandrov inequalities in a more familiar form.

Subfields: Applied Mathematics, Numerical Analysis

View Paper: https://www.semanticscholar.org/paper/ec110829a3b913d40db19d365aecad3e95c7cfb5

How researchers see this paper

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In Applied Mathematics

1. Andrea Colesanti
2. F. Mussnig
3. Monika Ludwig
4. Ilaria Fragalà
5. Sergey G. Bobkov
6. Franz E. Schuster
7. Emanuel Milman
8. Yiming Zhao

In Numerical Analysis

1. Andrea Colesanti
2. Monika Ludwig
3. Ilaria Fragalà
4. Sergey G. Bobkov
5. Emanuel Milman
6. Maxim Raginsky

Citations in the corpus

• A homogeneous decomposition theorem for valuations on convex functions (2019, background): “Milman and Rotem [31] discuss the problem to find a functional analog of Minkowski’s mixed volume theorem.”
• Quermassintegrals of quasi-concave functions and generalized Prékopa–Leindler inequalities (2012, result): “Rotem about the paper [25], where the authors present ideas and results, found independently, which partially overlap with those of the present paper.”
• Valuations on Convex Functions (2017, background): “Moreover, classical functionals for convex sets including the intrinsic volumes have been extended to the space of quasi-concave functions in [7] and [28] (see also [9, 18]).”