Winning tactics in a geometrical game
HTML articles powered by AMS MathViewer
- by Antonín Procházka PDF
- Proc. Amer. Math. Soc. 137 (2009), 1051-1061 Request permission
Abstract:
A winning tactic for the point-closed slice game in a closed bounded convex set $K$ with Radon-Nikodým property (RNP) is constructed. Consequently a Banach space $X$ has the RNP if and only if there exists a winning tactic in the point-closed slice game played in the unit ball of $X$. By contrast, there is no winning tactic in the point-open slice game in $K$. Finally, a more subtle analysis of the properties of the winning tactics leads to a characterization of superreflexive spaces.References
- Richard D. Bourgin, Geometric aspects of convex sets with the Radon-Nikodým property, Lecture Notes in Mathematics, vol. 993, Springer-Verlag, Berlin, 1983. MR 704815, DOI 10.1007/BFb0069321
- Robert Deville, Gilles Godefroy, and Václav Zizler, Smoothness and renormings in Banach spaces, Pitman Monographs and Surveys in Pure and Applied Mathematics, vol. 64, Longman Scientific & Technical, Harlow; copublished in the United States with John Wiley & Sons, Inc., New York, 1993. MR 1211634
- R. Deville ; J. Jaramillo, Almost classical solutions of Hamilton-Jacobi Equations, to appear in Rev. Mat. Iberoamericana.
- R. Deville and É. Matheron, Infinite games, Banach space geometry and the eikonal equation, Proc. Lond. Math. Soc. (3) 95 (2007), no. 1, 49–68. MR 2329548, DOI 10.1112/plms/pdm005
- Robert C. James, Some self-dual properties of normed linear spaces, Symposium on Infinite-Dimensional Topology (Louisiana State Univ., Baton Rouge, La., 1967) Ann. of Math. Studies, No. 69, Princeton Univ. Press, Princeton, N.J., 1972, pp. 159–175. MR 0454600
- J. Malý and M. Zelený, A note on Buczolich’s solution of the Weil gradient problem: a construction based on an infinite game, Acta Math. Hungar. 113 (2006), no. 1-2, 145–158. MR 2271458, DOI 10.1007/s10474-006-0096-7
- J. Munkres, Topology (2nd Edition) (Prentice Hall, Upper Saddle River, NJ, 2000).
- Robert R. Phelps, Convex functions, monotone operators and differentiability, 2nd ed., Lecture Notes in Mathematics, vol. 1364, Springer-Verlag, Berlin, 1993. MR 1238715
- M. Zelený, The Denjoy-Clarkson property with respect to Hausdorff measures for the gradient mapping of functions of several variables, to appear in Ann. Inst. Fourier (Grenoble).
Additional Information
- Antonín Procházka
- Affiliation: KMA MFF UK, Charles University, Sokolovská 83, 18675 Prague, Czech Republic
- Address at time of publication: Université Bordeaux 1, 351 cours de la Libération, 33405 Talence cedex, France
- Email: protony@math.u-bordeaux1.fr
- Received by editor(s): February 18, 2008
- Published electronically: September 26, 2008
- Additional Notes: The author was supported by the grant GA CR 201/07/0394.
- Communicated by: Nigel J. Kalton
- © Copyright 2008
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc. 137 (2009), 1051-1061
- MSC (2000): Primary 91A05, 46B20, 46B22; Secondary 47H04
- DOI: https://doi.org/10.1090/S0002-9939-08-09636-6
- MathSciNet review: 2457446