New Convexity and Fixed Point Properties in Hardy and Lebesgue-Bochner Spaces

We show that for the Hardy class of functions H1 with domain the ball or polydisc in CN, a certain type of uniform convexity property (the uniform Kadec-Klee-Huff property) holds with respect to the topology of pointwise convergence on the interior, which coincides with both the topology of uniform convergence on compacta and the weak* topology on bounded subsets of H1. Also, we show that if a Banach space X has a uniform Kadec-Klee-Huff property, then the Lebesgue-Bochner space Lp(μ, X), 1 ≤ p < ∞, must have a related uniform Kadec-Klee-Huff property. Consequently, by known results, the above spaces have normal structure properties and fixed point properties for non-expansive mappings.