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# Banach limit in convexity and geometric means for convex bodies

• 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.
Mathematics Subject Classification: 52A20, 46A22.

 Citation:

•  [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. [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. [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. [4] J. Lawson and Y. Lim, The geometric mean, matrices, metrics, and more, Amer. Math. Monthly, 108 (2001), 797-812.doi: 10.2307/2695553. [5] P. D. Lax, Functional Analysis, Pure and Applied Mathematics (New York), Wiley-Interscience [John Wiley & Sons], New York, 2002. [6] V. Milman and L. Rotem, Non-standard constructions in convex geometry; geometric means of convex bodies, in Convexity, Concentration and Discrete Structures (eds. E. Carlen, M. Madiman and E. Werner), The IMA Volumes in Mathematics and its Applications (to appear), Springer. [7] I. Molchanov, Continued fractions built from convex sets and convex functions, Commun. Contemp. Math., 17 (2015), 1550003, 18 pp.doi: 10.1142/S0219199715500030. [8] L. Rotem, Algebraically inspired results on convex functions and bodies, Commun. Contemp. Math., 18 (2016), 1650027, 14 pp.doi: 10.1142/S0219199716500279. [9] R. Schneider, Convex Bodies: The Brunn-Minkowski Theory, Second expanded edition, Encyclopedia of Mathematics and its Applications, 151, Cambridge University Press, Cambridge, 2014.

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