# American Institute of Mathematical Sciences

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

## $\alpha$-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

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.
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.
 [1] Michael Kühn. Power- and Log-concavity of viscosity solutions to some elliptic Dirichlet problems. Communications on Pure and Applied Analysis, 2018, 17 (6) : 2773-2788. doi: 10.3934/cpaa.2018131 [2] Yao Kuang, Raphael Douady. Crisis risk prediction with concavity from Polymodel. Journal of Dynamics and Games, 2022, 9 (1) : 97-115. doi: 10.3934/jdg.2021027 [3] Alexander J. Zaslavski. Good programs in the RSS model without concavity of a utility function. Journal of Industrial and Management Optimization, 2006, 2 (4) : 399-423. doi: 10.3934/jimo.2006.2.399 [4] Laura Scrimali. Mixed behavior network equilibria and quasi-variational inequalities. Journal of Industrial and Management Optimization, 2009, 5 (2) : 363-379. doi: 10.3934/jimo.2009.5.363 [5] Sara Munday. On the derivative of the $\alpha$-Farey-Minkowski function. Discrete and Continuous Dynamical Systems, 2014, 34 (2) : 709-732. doi: 10.3934/dcds.2014.34.709 [6] Murat Adivar, Shu-Cherng Fang. Convex optimization on mixed domains. Journal of Industrial and Management Optimization, 2012, 8 (1) : 189-227. doi: 10.3934/jimo.2012.8.189 [7] Meijuan Shang, Yanan Liu, Lingchen Kong, Xianchao Xiu, Ying Yang. Nonconvex mixed matrix minimization. Mathematical Foundations of Computing, 2019, 2 (2) : 107-126. doi: 10.3934/mfc.2019009 [8] Fredrik Hellman, Patrick Henning, Axel Målqvist. Multiscale mixed finite elements. Discrete and Continuous Dynamical Systems - S, 2016, 9 (5) : 1269-1298. doi: 10.3934/dcdss.2016051 [9] Eliane Bécache, Laurent Bourgeois, Lucas Franceschini, Jérémi Dardé. Application of mixed formulations of quasi-reversibility to solve ill-posed problems for heat and wave equations: The 1D case. Inverse Problems and Imaging, 2015, 9 (4) : 971-1002. doi: 10.3934/ipi.2015.9.971 [10] Ferdinando Auricchio, Lourenco Beirão da Veiga, Josef Kiendl, Carlo Lovadina, Alessandro Reali. Isogeometric collocation mixed methods for rods. Discrete and Continuous Dynamical Systems - S, 2016, 9 (1) : 33-42. doi: 10.3934/dcdss.2016.9.33 [11] Chiu-Yen Kao, Yuan Lou, Wenxian Shen. Evolution of mixed dispersal in periodic environments. Discrete and Continuous Dynamical Systems - B, 2012, 17 (6) : 2047-2072. doi: 10.3934/dcdsb.2012.17.2047 [12] Hsin-Lun Li. Mixed Hegselmann-Krause dynamics. Discrete and Continuous Dynamical Systems - B, 2022, 27 (2) : 1149-1162. doi: 10.3934/dcdsb.2021084 [13] Salahuddin. System of generalized mixed nonlinear ordered variational inclusions. Numerical Algebra, Control and Optimization, 2019, 9 (4) : 445-460. doi: 10.3934/naco.2019026 [14] Hongyu Liu, Jun Zou. Uniqueness in determining multiple polygonal scatterers of mixed type. Discrete and Continuous Dynamical Systems - B, 2008, 9 (2) : 375-396. doi: 10.3934/dcdsb.2008.9.375 [15] J. K. Krottje. On the dynamics of a mixed parabolic-gradient system. Communications on Pure and Applied Analysis, 2003, 2 (4) : 521-537. doi: 10.3934/cpaa.2003.2.521 [16] Antoine Benoit. Finite speed of propagation for mixed problems in the $WR$ class. Communications on Pure and Applied Analysis, 2014, 13 (6) : 2351-2358. doi: 10.3934/cpaa.2014.13.2351 [17] T. Candan, R.S. Dahiya. Oscillation of mixed neutral differential equations with forcing term. Conference Publications, 2003, 2003 (Special) : 167-172. doi: 10.3934/proc.2003.2003.167 [18] Cesare Tronci. Momentum maps for mixed states in quantum and classical mechanics. Journal of Geometric Mechanics, 2019, 11 (4) : 639-656. doi: 10.3934/jgm.2019032 [19] Thomas Demoor, Dieter Fiems, Joris Walraevens, Herwig Bruneel. Partially shared buffers with full or mixed priority. Journal of Industrial and Management Optimization, 2011, 7 (3) : 735-751. doi: 10.3934/jimo.2011.7.735 [20] Astridh Boccabella, Roberto Natalini, Lorenzo Pareschi. On a continuous mixed strategies model for evolutionary game theory. Kinetic and Related Models, 2011, 4 (1) : 187-213. doi: 10.3934/krm.2011.4.187

2020 Impact Factor: 0.929