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$\alpha$-concave functions and a functional extension of mixed volumes
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 |
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.
References:
[1] |
Mordecai Avriel, r-convex functions,, Mathematical Programming, 2 (1972), 309.
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[2] |
Sergey Bobkov, Convex bodies and norms associated to convex measures,, Probability Theory and Related Fields, 147 (2009), 303.
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), (2012). Google Scholar |
[4] |
Christer Borell, Convex measures on locally convex spaces,, Arkiv för Matematik, 12 (1974), 239.
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[5] |
Christer Borell, Convex set functions in d-space,, Periodica Mathematica Hungarica, 6 (1975), 111.
|
[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.
|
[7] |
Bo'az Klartag and Vitali Milman, Geometry of log-concave functions and measures,, Geometriae Dedicata, 112 (2005), 169.
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.
doi: 10.1016/j.jfa.2012.10.019. |
[9] |
Liran Rotem, Support functions and mean width for $\alpha$-concave functions,, preprint, (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 (1993).
doi: 10.1017/CBO9780511526282. |
show all references
References:
[1] |
Mordecai Avriel, r-convex functions,, Mathematical Programming, 2 (1972), 309.
|
[2] |
Sergey Bobkov, Convex bodies and norms associated to convex measures,, Probability Theory and Related Fields, 147 (2009), 303.
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), (2012). Google Scholar |
[4] |
Christer Borell, Convex measures on locally convex spaces,, Arkiv för Matematik, 12 (1974), 239.
|
[5] |
Christer Borell, Convex set functions in d-space,, Periodica Mathematica Hungarica, 6 (1975), 111.
|
[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.
|
[7] |
Bo'az Klartag and Vitali Milman, Geometry of log-concave functions and measures,, Geometriae Dedicata, 112 (2005), 169.
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.
doi: 10.1016/j.jfa.2012.10.019. |
[9] |
Liran Rotem, Support functions and mean width for $\alpha$-concave functions,, preprint, (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 (1993).
doi: 10.1017/CBO9780511526282. |
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