Multi-dimensional volumes and moduli of convexity in banach spaces

Summary Let E be a Banach space. Using the definition for the k-dimensional volume enclosed by k + 1 vectors due to Silverman 16], one can define the modulus of k-rotundity of E. In 22] it was shown that k-uniformly rotund Banach spaces are isomorphic to uniformly rotund spaces and, indeed, have some of the same isometric properties with respect to non-expansive and nearest-point maps. The present paper examines the modulus of k-rotundity more thoroughly. Included are a result on the asymptotic behavior of the moduli for l²; a generalization of Dixmier's Theorem on higher-duals of non-reflexive spaces; and an inequality relating the moduli of E^**/E and those of E. The modulus of2-rotundity is shown to be equivalent to one of the moduli defined by V. D. Milman 13] and a necessary and sufficient condition for an l^p-product of spaces to be 2-uniformly rotund is given.Some of the results of this paper are contained in the Ph. D. dissertation of the first author, written under the direction of the second author.