American Institute of Mathematical Sciences

Advanced
  2016, 23: 41-51. doi: 10.3934/era.2016.23.005

Banach limit in convexity and geometric means for convex bodies

Liran Rotem 1,

1. 

University of Minnesota, School of Mathematics, United States

Received  August 2016 Revised  October 2016 Published  November 2016

In this note we construct Banach limits on the class of sequences of convex bodies. Surprisingly, the construction uses the recently introduced geometric mean of convex bodies. In the opposite direction, we explain how Banach limits can be used to construct a new variant of the geometric mean that has some desirable properties.
Keywords: geometric mean, Banach limit., Convex bodies.
Mathematics Subject Classification: 52A20, 46A2.
Citation: Liran Rotem. Banach limit in convexity and geometric means for convex bodies. Electronic Research Announcements, 2016, 23: 41-51. doi: 10.3934/era.2016.23.005
References:
[1]

S. Artstein-Avidan and V. Milman, The concept of duality for measure projections of convex bodies,, J. Funct. Anal., 254 (2008), 2648.  doi: 10.1016/j.jfa.2007.11.008.  Google Scholar

[2]

K. J. Böröczky and R. Schneider, A characterization of the duality mapping for convex bodies,, Geom. Funct. Anal., 18 (2008), 657.  doi: 10.1007/s00039-008-0676-5.  Google Scholar

[3]

P. M. Gruber, The endomorphisms of the lattice of norms in finite dimensions,, Abh. Math. Sem. Univ. Hamburg, 62 (1992), 179.  doi: 10.1007/BF02941625.  Google Scholar

[4]

J. Lawson and Y. Lim, The geometric mean, matrices, metrics, and more,, Amer. Math. Monthly, 108 (2001), 797.  doi: 10.2307/2695553.  Google Scholar

[5]

P. D. Lax, Functional Analysis,, Pure and Applied Mathematics (New York), (2002).   Google Scholar

[6]

V. Milman and L. Rotem, Non-standard constructions in convex geometry; geometric means of convex bodies,, in Convexity, ().   Google Scholar

[7]

I. Molchanov, Continued fractions built from convex sets and convex functions,, Commun. Contemp. Math., 17 (2015).  doi: 10.1142/S0219199715500030.  Google Scholar

[8]

L. Rotem, Algebraically inspired results on convex functions and bodies,, Commun. Contemp. Math., 18 (2016).  doi: 10.1142/S0219199716500279.  Google Scholar

[9]

R. Schneider, Convex Bodies: The Brunn-Minkowski Theory,, Second expanded edition, (2014).   Google Scholar

show all references

References:
[1]

S. Artstein-Avidan and V. Milman, The concept of duality for measure projections of convex bodies,, J. Funct. Anal., 254 (2008), 2648.  doi: 10.1016/j.jfa.2007.11.008.  Google Scholar

[2]

K. J. Böröczky and R. Schneider, A characterization of the duality mapping for convex bodies,, Geom. Funct. Anal., 18 (2008), 657.  doi: 10.1007/s00039-008-0676-5.  Google Scholar

[3]

P. M. Gruber, The endomorphisms of the lattice of norms in finite dimensions,, Abh. Math. Sem. Univ. Hamburg, 62 (1992), 179.  doi: 10.1007/BF02941625.  Google Scholar

[4]

J. Lawson and Y. Lim, The geometric mean, matrices, metrics, and more,, Amer. Math. Monthly, 108 (2001), 797.  doi: 10.2307/2695553.  Google Scholar

[5]

P. D. Lax, Functional Analysis,, Pure and Applied Mathematics (New York), (2002).   Google Scholar

[6]

V. Milman and L. Rotem, Non-standard constructions in convex geometry; geometric means of convex bodies,, in Convexity, ().   Google Scholar

[7]

I. Molchanov, Continued fractions built from convex sets and convex functions,, Commun. Contemp. Math., 17 (2015).  doi: 10.1142/S0219199715500030.  Google Scholar

[8]

L. Rotem, Algebraically inspired results on convex functions and bodies,, Commun. Contemp. Math., 18 (2016).  doi: 10.1142/S0219199716500279.  Google Scholar

[9]

R. Schneider, Convex Bodies: The Brunn-Minkowski Theory,, Second expanded edition, (2014).   Google Scholar

[1]

Franco Flandoli, Matti Leimbach. Mean field limit with proliferation. Discrete & Continuous Dynamical Systems - B, 2016, 21 (9) : 3029-3052. doi: 10.3934/dcdsb.2016086
[2]

Tobias H. Colding and Bruce Kleiner. Singularity structure in mean curvature flow of mean-convex sets. Electronic Research Announcements, 2003, 9: 121-124.

[3]

Lei Wu. Diffusive limit with geometric correction of unsteady neutron transport equation. Kinetic & Related Models, 2017, 10 (4) : 1163-1203. doi: 10.3934/krm.2017045
[4]

Samir Adly, Ba Khiet Le. Unbounded state-dependent sweeping processes with perturbations in uniformly convex and q-uniformly smooth Banach spaces. Numerical Algebra, Control & Optimization, 2018, 8 (1) : 81-95. doi: 10.3934/naco.2018005
[5]

Gerasimenko Viktor. Heisenberg picture of quantum kinetic evolution in mean-field limit. Kinetic & Related Models, 2011, 4 (1) : 385-399. doi: 10.3934/krm.2011.4.385
[6]

Seung-Yeal Ha, Jeongho Kim, Jinyeong Park, Xiongtao Zhang. Uniform stability and mean-field limit for the augmented Kuramoto model. Networks & Heterogeneous Media, 2018, 13 (2) : 297-322. doi: 10.3934/nhm.2018013
[7]

Michael Herty, Mattia Zanella. Performance bounds for the mean-field limit of constrained dynamics. Discrete & Continuous Dynamical Systems - A, 2017, 37 (4) : 2023-2043. doi: 10.3934/dcds.2017086
[8]

Juan Pablo Maldonado López. Discrete time mean field games: The short-stage limit. Journal of Dynamics & Games, 2015, 2 (1) : 89-101. doi: 10.3934/jdg.2015.2.89
[9]

Franco Flandoli, Marta Leocata, Cristiano Ricci. The Vlasov-Navier-Stokes equations as a mean field limit. Discrete & Continuous Dynamical Systems - B, 2019, 24 (8) : 3741-3753. doi: 10.3934/dcdsb.2018313
[10]

Seung-Yeal Ha, Jeongho Kim, Xiongtao Zhang. Uniform stability of the Cucker-Smale model and its application to the Mean-Field limit. Kinetic & Related Models, 2018, 11 (5) : 1157-1181. doi: 10.3934/krm.2018045
[11]

Kazuhisa Ichikawa, Mahemauti Rouzimaimaiti, Takashi Suzuki. Reaction diffusion equation with non-local term arises as a mean field limit of the master equation. Discrete & Continuous Dynamical Systems - S, 2012, 5 (1) : 115-126. doi: 10.3934/dcdss.2012.5.115
[12]

Rong Yang, Li Chen. Mean-field limit for a collision-avoiding flocking system and the time-asymptotic flocking dynamics for the kinetic equation. Kinetic & Related Models, 2014, 7 (2) : 381-400. doi: 10.3934/krm.2014.7.381
[13]

Young-Pil Choi, Samir Salem. Cucker-Smale flocking particles with multiplicative noises: Stochastic mean-field limit and phase transition. Kinetic & Related Models, 2019, 12 (3) : 573-592. doi: 10.3934/krm.2019023
[14]

Seung-Yeal Ha, Jeongho Kim, Peter Pickl, Xiongtao Zhang. A probabilistic approach for the mean-field limit to the Cucker-Smale model with a singular communication. Kinetic & Related Models, 2019, 12 (5) : 1045-1067. doi: 10.3934/krm.2019039
[15]

Joachim Crevat. Mean-field limit of a spatially-extended FitzHugh-Nagumo neural network. Kinetic & Related Models, 2019, 12 (6) : 1329-1358. doi: 10.3934/krm.2019052
[16]

Patricia Gaitan, Hiroshi Isozaki, Olivier Poisson, Samuli Siltanen, Janne Tamminen. Probing for inclusions in heat conductive bodies. Inverse Problems & Imaging, 2012, 6 (3) : 423-446. doi: 10.3934/ipi.2012.6.423
[17]

Jean-Marie Souriau. On Geometric Mechanics. Discrete & Continuous Dynamical Systems - A, 2007, 19 (3) : 595-607. doi: 10.3934/dcds.2007.19.595
[18]

Sonja Cox, Arnulf Jentzen, Ryan Kurniawan, Primož Pušnik. On the mild Itô formula in Banach spaces. Discrete & Continuous Dynamical Systems - B, 2018, 23 (6) : 2217-2243. doi: 10.3934/dcdsb.2018232
[19]

Alexander Plakhov, Vera Roshchina. Fractal bodies invisible in 2 and 3 directions. Discrete & Continuous Dynamical Systems - A, 2013, 33 (4) : 1615-1631. doi: 10.3934/dcds.2013.33.1615
[20]

Joris Vankerschaver, Eva Kanso, Jerrold E. Marsden. The geometry and dynamics of interacting rigid bodies and point vortices. Journal of Geometric Mechanics, 2009, 1 (2) : 223-266. doi: 10.3934/jgm.2009.1.223

2018 Impact Factor: 0.263

Tools

Metrics

  • PDF downloads (28)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences