Publication: A constant of porosity for convex bodies
Full text at PDC
Advisors (or tutors)
University of Illinois
It was proved recently that a Banach space fails the Mazur intersection property if and only if the family of all closed, convex and bounded subsets which are intersections of balls is uniformly very porous. This paper deals with the geometrical implications of this result. It is shown that every equivalent norm on the space can be associated in a natural way with a constant of porosity, whose interplay with the geometry of the space is then investigated. Among other things, we prove that this constant is closely related to the set of ε-differentiability points of the space and the set of r-denting points of the dual. We also obtain estimates for this constant in several classical spaces.
Supported in part by DGICYT Grant BMF-2000-0609.The authors wish to thank the C.E.C.M., the Department of Mathematics and Statistics at Simon Fraser University, and especially J. Borwein, for their hospitality during the preparation of this paper.
G. Beer, Topologies on closed and closed convex sets, Mathematics and its Applications, vol. 268, Kluwer Academic Publishers, Dordrecht, 1993. D. Chen and B.-L. Lin, Ball separation properties in Banach spaces, Rocky Mountain J. Math. 28 (1998), 835–873. R. Deville, G. Godefroy, and V. Zizler, Smoothness and renormings in Banach spaces, Pitman Monographs and Surveys in Pure and Applied Mathematics, vol. 64, Longman Scientific, Harlow, 1993. P. G. Georgiev, On the residuality of the set of norms having Mazur's intersection property, Math. Balkanica, 5 (1991), 20–26. J. R. Giles, D. A. Gregory, and B. Sims, Characterization of normed linear spaces with Mazur's intersection property, Bull. Austral. Math. Soc. 18 (1978), 471–476. P. M. Gruber, Baire categories in convexity, Handbook of convex geometry (P. M. Gruber and J. M. Wills eds.), North-Holland, 1993, pp. 1327–1346. M. Jiménez Sevilla and J.P. Moreno, Renorming Banach spaces with the Mazur intersection property, J. Funct. Anal. 144 (1997) 486–504. M. Jiménez Sevilla, J.P. Moreno, A note on porosity and the Mazur intersection property, Mathematika, to appear. J. Lindenstrauss, On operators which attain their norm, Israel J. Math. 1 (1963) 139–148. S. Mazur, Über schwache Konvergenz in den Raumen L p , Studia Math. 4 (1933), 128–133. J. P. Moreno, Geometry of Banach spaces with (α,ε) -property or (β,ε) -property, Rocky Mountain J. Math. 27 (1997), 241–256. J. R. Partington, Norm attaining operators, Israel J. Math. 1 (1983), 273–276. R. R. Phelps, Convex functions, monotone operators and differentiability. Second edition, Lecture Notes in Math., vol. 1364, Springer Verlag, Berlin, 1993. R. R. Phelps, A representation theorem for bounded convex sets, Proc. Amer. Math. Soc. 11 (1960), 976–983. S.W. Schachermayer, Norm attaining operators and renormings of Banach spaces, Israel J. Math. 44 (1983), 201–212. F. Sullivan, Dentability, smoothability and stronger properties in Banach spaces, Indiana Math. J. 26 (1977), 545–553. L. Zajicek, Porosity and σ -porosity, Real Analysis Exchange 13 (1987-88), 314–350. T. Zamfirescu, Porosity in convexity, Real Analysis Exchange 15 (1989-90), 424–436. T. Zamfirescu, Baire categories in convexity, Atti Sem. Mat. Fis. Univ. Modena 39 (1991), 139–164.