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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

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
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show all references

References:
[1]

Mathematical Programming, 2 (1972), 309-323.  Google Scholar

[2]

Probability Theory and Related Fields, 147 (2009), 303-332. doi: 10.1007/s00440-009-0209-7.  Google Scholar

[3]

(2012), 36 pp. Google Scholar

[4]

Arkiv för Matematik, 12 (1974), 239-252.  Google Scholar

[5]

Periodica Mathematica Hungarica, 6 (1975), 111-136.  Google Scholar

[6]

Journal of Functional Analysis, 22 (1976), 366-389.  Google Scholar

[7]

Geometriae Dedicata, 112 (2005), 169-182. doi: 10.1007/s10711-004-2462-3.  Google Scholar

[8]

Journal of Functional Analysis, 264 (2013), 570-604. doi: 10.1016/j.jfa.2012.10.019.  Google Scholar

[9]

preprint, arXiv:1210.4340, (2012). doi: 10.1016/j.bulsci.2012.03.003.  Google Scholar

[10]

Encyclopedia of Mathematics and its Applications, 44, Cambridge University Press, Cambridge, 1993. doi: 10.1017/CBO9780511526282.  Google Scholar

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