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where 11 * 112 and 11 IjX denote the 12 and lx norms, respectively. It is known from [1] that Er has normal structure when r < V2. We use the idea of multidimensional volumes and their related convexity moduli to prove Er satisfies a convexity property that implies this result. The notion of volumes in Banach spaces was introduced by Silverman and its use in defining moduli of convexity was introduced in [5]. Roughly speaking, the modulus of k-rotundity, Sk(f), measures the depth below the surface of the unit sphere of the centroid of a simplex of k + 1 norm-i vectors enclosing a k-dimensional volume larger than c. In symbols,

Banach spaces that have normal structure and are isomorphic to a Hilbert space