\alpha-concave functions and a functional extension of mixed volumes

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2013, 20: 1-11. doi: 10.3934/era.2013.20.1

$\alpha$-concave functions and a functional extension of mixed volumes

Vitali Milman ^1, and Liran Rotem ^2,

1.	School of Mathematical Sciences, Tel Aviv University, Ramat Aviv, Tel Aviv 69978
2.	School of Mathematical Sciences, Tel Aviv University, Tel Aviv 69978, Israel

Received November 2012 Revised December 2012 Published January 2013

Abstract
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Mixed volumes, which are the polarization of volume with respect to the Minkowski addition, are fundamental objects in convexity. In this note we announce the construction of mixed integrals, which are functional analogs of mixed volumes. We build a natural addition operation $\oplus$ on the class of quasi-concave functions, such that every class of $\alpha$-concave functions is closed under $\oplus$. We then define the mixed integrals, which are the polarization of the integral with respect to $\oplus$.
We proceed to discuss the extension of various classic 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 $\alpha$-concave functions, we state a generalization of the Alexandrov inequalities in their more familiar form.

Keywords: alpha-concavity, log-concavity, Mixed integrals, Brunn-Minkowski., mixed volumes, quasi-concavity.

Mathematics Subject Classification: Primary: 52A39; Secondary: 26B2.

Citation: Vitali Milman, Liran Rotem. $\alpha$-concave functions and a functional extension of mixed volumes. Electronic Research Announcements, 2013, 20: 1-11. doi: 10.3934/era.2013.20.1

References:

[1]	Mordecai Avriel, r-convex functions, Mathematical Programming, 2 (1972), 309-323.
[2]	Sergey Bobkov, Convex bodies and norms associated to convex measures, Probability Theory and Related Fields, 147 (2009), 303-332. doi: 10.1007/s00440-009-0209-7.
[3]	Sergey Bobkov, Andrea Colesanti and Ilaria Fragalà, Quermassintegrals of quasi-concave functions and generalized Prékopa-Leindler inequalities, (2012), 36 pp.
[4]	Christer Borell, Convex measures on locally convex spaces, Arkiv för Matematik, 12 (1974), 239-252.
[5]	Christer Borell, Convex set functions in d-space, Periodica Mathematica Hungarica, 6 (1975), 111-136.
[6]	Herm J. Brascamp and Elliott H. Lieb, On extensions of the Brunn-Minkowski and Prékopa-Leindler theorems, including inequalities for log concave functions, and with an application to the diffusion equation, Journal of Functional Analysis, 22 (1976), 366-389.
[7]	Bo'az Klartag and Vitali Milman, Geometry of log-concave functions and measures, Geometriae Dedicata, 112 (2005), 169-182. doi: 10.1007/s10711-004-2462-3.
[8]	Vitali Milman and Liran Rotem, Mixed integrals and related inequalities, Journal of Functional Analysis, 264 (2013), 570-604. doi: 10.1016/j.jfa.2012.10.019.
[9]	Liran Rotem, Support functions and mean width for $\alpha$-concave functions, preprint, arXiv:1210.4340, (2012). doi: 10.1016/j.bulsci.2012.03.003.
[10]	Rolf Schneider, "Convex Bodies: The Brunn-Minkowski Theory," Encyclopedia of Mathematics and its Applications, 44, Cambridge University Press, Cambridge, 1993. doi: 10.1017/CBO9780511526282.

show all references

References:

[1]	Mordecai Avriel, r-convex functions, Mathematical Programming, 2 (1972), 309-323.
[2]	Sergey Bobkov, Convex bodies and norms associated to convex measures, Probability Theory and Related Fields, 147 (2009), 303-332. doi: 10.1007/s00440-009-0209-7.
[3]	Sergey Bobkov, Andrea Colesanti and Ilaria Fragalà, Quermassintegrals of quasi-concave functions and generalized Prékopa-Leindler inequalities, (2012), 36 pp.
[4]	Christer Borell, Convex measures on locally convex spaces, Arkiv för Matematik, 12 (1974), 239-252.
[5]	Christer Borell, Convex set functions in d-space, Periodica Mathematica Hungarica, 6 (1975), 111-136.
[6]	Herm J. Brascamp and Elliott H. Lieb, On extensions of the Brunn-Minkowski and Prékopa-Leindler theorems, including inequalities for log concave functions, and with an application to the diffusion equation, Journal of Functional Analysis, 22 (1976), 366-389.
[7]	Bo'az Klartag and Vitali Milman, Geometry of log-concave functions and measures, Geometriae Dedicata, 112 (2005), 169-182. doi: 10.1007/s10711-004-2462-3.
[8]	Vitali Milman and Liran Rotem, Mixed integrals and related inequalities, Journal of Functional Analysis, 264 (2013), 570-604. doi: 10.1016/j.jfa.2012.10.019.
[9]	Liran Rotem, Support functions and mean width for $\alpha$-concave functions, preprint, arXiv:1210.4340, (2012). doi: 10.1016/j.bulsci.2012.03.003.
[10]	Rolf Schneider, "Convex Bodies: The Brunn-Minkowski Theory," Encyclopedia of Mathematics and its Applications, 44, Cambridge University Press, Cambridge, 1993. doi: 10.1017/CBO9780511526282.

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