# American Institute of Mathematical Sciences

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

## Banach limit in convexity and geometric means for convex bodies

 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.
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), 2648-2666. 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), 657-667. 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), 179-189. doi: 10.1007/BF02941625.  Google Scholar [4] J. Lawson and Y. Lim, The geometric mean, matrices, metrics, and more, Amer. Math. Monthly, 108 (2001), 797-812. doi: 10.2307/2695553.  Google Scholar [5] P. D. Lax, Functional Analysis, Pure and Applied Mathematics (New York), Wiley-Interscience [John Wiley & Sons], 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), 1550003, 18 pp. doi: 10.1142/S0219199715500030.  Google Scholar [8] L. Rotem, Algebraically inspired results on convex functions and bodies, Commun. Contemp. Math., 18 (2016), 1650027, 14 pp. doi: 10.1142/S0219199716500279.  Google Scholar [9] R. Schneider, Convex Bodies: The Brunn-Minkowski Theory, Second expanded edition, Encyclopedia of Mathematics and its Applications, 151, Cambridge University Press, Cambridge, 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), 2648-2666. 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), 657-667. 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), 179-189. doi: 10.1007/BF02941625.  Google Scholar [4] J. Lawson and Y. Lim, The geometric mean, matrices, metrics, and more, Amer. Math. Monthly, 108 (2001), 797-812. doi: 10.2307/2695553.  Google Scholar [5] P. D. Lax, Functional Analysis, Pure and Applied Mathematics (New York), Wiley-Interscience [John Wiley & Sons], 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), 1550003, 18 pp. doi: 10.1142/S0219199715500030.  Google Scholar [8] L. Rotem, Algebraically inspired results on convex functions and bodies, Commun. Contemp. Math., 18 (2016), 1650027, 14 pp. doi: 10.1142/S0219199716500279.  Google Scholar [9] R. Schneider, Convex Bodies: The Brunn-Minkowski Theory, Second expanded edition, Encyclopedia of Mathematics and its Applications, 151, Cambridge University Press, Cambridge, 2014.  Google Scholar
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