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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
References:
[1]

Mordecai Avriel, r-convex functions, Mathematical Programming, 2 (1972), 309-323.  Google Scholar

[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.  Google Scholar

[3]

Sergey Bobkov, Andrea Colesanti and Ilaria Fragalà, Quermassintegrals of quasi-concave functions and generalized Prékopa-Leindler inequalities, (2012), 36 pp. Google Scholar

[4]

Christer Borell, Convex measures on locally convex spaces, Arkiv för Matematik, 12 (1974), 239-252.  Google Scholar

[5]

Christer Borell, Convex set functions in d-space, Periodica Mathematica Hungarica, 6 (1975), 111-136.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

show all references

References:
[1]

Mordecai Avriel, r-convex functions, Mathematical Programming, 2 (1972), 309-323.  Google Scholar

[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.  Google Scholar

[3]

Sergey Bobkov, Andrea Colesanti and Ilaria Fragalà, Quermassintegrals of quasi-concave functions and generalized Prékopa-Leindler inequalities, (2012), 36 pp. Google Scholar

[4]

Christer Borell, Convex measures on locally convex spaces, Arkiv för Matematik, 12 (1974), 239-252.  Google Scholar

[5]

Christer Borell, Convex set functions in d-space, Periodica Mathematica Hungarica, 6 (1975), 111-136.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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