American Institute of Mathematical Sciences

Advanced
  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.   Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[5]

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

[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.  Google Scholar

[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.  Google Scholar

[9]

Liran Rotem, Support functions and mean width for $\alpha$-concave functions,, preprint, (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 (1993).  doi: 10.1017/CBO9780511526282.  Google Scholar

show all references

References:
[1]

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

[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.  Google Scholar

[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.   Google Scholar

[5]

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

[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.  Google Scholar

[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.  Google Scholar

[9]

Liran Rotem, Support functions and mean width for $\alpha$-concave functions,, preprint, (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 (1993).  doi: 10.1017/CBO9780511526282.  Google Scholar

[1]

Michael Kühn. Power- and Log-concavity of viscosity solutions to some elliptic Dirichlet problems. Communications on Pure & Applied Analysis, 2018, 17 (6) : 2773-2788. doi: 10.3934/cpaa.2018131
[2]

Alexander J. Zaslavski. Good programs in the RSS model without concavity of a utility function. Journal of Industrial & Management Optimization, 2006, 2 (4) : 399-423. doi: 10.3934/jimo.2006.2.399
[3]

Laura Scrimali. Mixed behavior network equilibria and quasi-variational inequalities. Journal of Industrial & Management Optimization, 2009, 5 (2) : 363-379. doi: 10.3934/jimo.2009.5.363
[4]

Sara Munday. On the derivative of the $\alpha$-Farey-Minkowski function. Discrete & Continuous Dynamical Systems - A, 2014, 34 (2) : 709-732. doi: 10.3934/dcds.2014.34.709
[5]

Murat Adivar, Shu-Cherng Fang. Convex optimization on mixed domains. Journal of Industrial & Management Optimization, 2012, 8 (1) : 189-227. doi: 10.3934/jimo.2012.8.189
[6]

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
[7]

Fredrik Hellman, Patrick Henning, Axel Målqvist. Multiscale mixed finite elements. Discrete & Continuous Dynamical Systems - S, 2016, 9 (5) : 1269-1298. doi: 10.3934/dcdss.2016051
[8]

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 & Imaging, 2015, 9 (4) : 971-1002. doi: 10.3934/ipi.2015.9.971
[9]

Ferdinando Auricchio, Lourenco Beirão da Veiga, Josef Kiendl, Carlo Lovadina, Alessandro Reali. Isogeometric collocation mixed methods for rods. Discrete & Continuous Dynamical Systems - S, 2016, 9 (1) : 33-42. doi: 10.3934/dcdss.2016.9.33
[10]

Chiu-Yen Kao, Yuan Lou, Wenxian Shen. Evolution of mixed dispersal in periodic environments. Discrete & Continuous Dynamical Systems - B, 2012, 17 (6) : 2047-2072. doi: 10.3934/dcdsb.2012.17.2047
[11]

Salahuddin. System of generalized mixed nonlinear ordered variational inclusions. Numerical Algebra, Control & Optimization, 2019, 9 (4) : 445-460. doi: 10.3934/naco.2019026
[12]

Hongyu Liu, Jun Zou. Uniqueness in determining multiple polygonal scatterers of mixed type. Discrete & Continuous Dynamical Systems - B, 2008, 9 (2) : 375-396. doi: 10.3934/dcdsb.2008.9.375
[13]

J. K. Krottje. On the dynamics of a mixed parabolic-gradient system. Communications on Pure & Applied Analysis, 2003, 2 (4) : 521-537. doi: 10.3934/cpaa.2003.2.521
[14]

Antoine Benoit. Finite speed of propagation for mixed problems in the $WR$ class. Communications on Pure & Applied Analysis, 2014, 13 (6) : 2351-2358. doi: 10.3934/cpaa.2014.13.2351
[15]

Thomas Demoor, Dieter Fiems, Joris Walraevens, Herwig Bruneel. Partially shared buffers with full or mixed priority. Journal of Industrial & Management Optimization, 2011, 7 (3) : 735-751. doi: 10.3934/jimo.2011.7.735
[16]

Astridh Boccabella, Roberto Natalini, Lorenzo Pareschi. On a continuous mixed strategies model for evolutionary game theory. Kinetic & Related Models, 2011, 4 (1) : 187-213. doi: 10.3934/krm.2011.4.187
[17]

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
[18]

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
[19]

Xiaoyu Fu. Stabilization of hyperbolic equations with mixed boundary conditions. Mathematical Control & Related Fields, 2015, 5 (4) : 761-780. doi: 10.3934/mcrf.2015.5.761
[20]

Martin Burger, José A. Carrillo, Marie-Therese Wolfram. A mixed finite element method for nonlinear diffusion equations. Kinetic & Related Models, 2010, 3 (1) : 59-83. doi: 10.3934/krm.2010.3.59

2019 Impact Factor: 0.5

Tools

Metrics

  • PDF downloads (36)
  • HTML views (0)
  • Cited by (3)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences