NEW CONVEXITY AND FIXED-POINT PROPERTIES IN HARDY AND LEBESGUE-BOCHNER SPACES

We show that for the Hardy class of functions H-1 with domain the ball or polydisc in C(N), a certain type of uniform convexity property (th e uniform Kadec-Klee-Huff property) holds with respect to the topology of pointwise convergence on the interior, which coincides with both t he topology of uniform convergence on compacta and the weak topology on bounded subsets of H-1. Also, we show that if a Banach space X has a uniform Kadec-Klee-Huff property, then the Lebesgue-Bochner space L( p)(mu, X), 1 less-than-or-equal-to p < infinity, must have a related u niform Kadec-Klee-Huff property. Consequently, by known results, the a bove spaces have normal structure properties and fixed point propertie s for non-expansive mappings. (C) 1994 Academic Press, Inc.