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Again identify both spaces X and Y with Rn or Cn . When one convex body contains another, a point x is called a contact point of the bodies if it is in the intersection of their boundaries (so if the bodies are unit balls associated with two norms, a contact point is a point in the intersection of the two unit spheres). Assume that BX ⊂ BY and vol (BX ) ≥ vol (T BX ) for every operator T on Rn [Cn ] for which T BX ⊂ BY . Then there exist contact points x1 , . . , xN of BX and BY and contact points x∗1 , .

For simplicity, we assume that X is a lattice of µ-measurable functions (for the general case see [15, p. 53]). For 1 < p < ∞, the p-convexification of X is the space X (p) of all µ-measurable functions x for which |x|p sign (x) is in X. This is easily seen to 1/p be a Banach lattice under the norm x X (p) := |x|p X . The space X (p) is p-convex with M (p) (X (p) ) = 1. More generally, if X is r-convex and s-concave, then X (p) is pr-convex and ps-concave with moduli M (pr) (X (p) ) ≤ M (r) (X)1/p and M(ps) (X (p) ) ≤ M(s) (X)1/p .

E. By replacing the measure µ with u dµ and functions f in X by f /u, one can assume u ≡ 1. Now it is possible to recover the µmeasurable sets and the measure µ. From the density of X in L1 (µ) it follows that the indicator function of a measurable set is an element x in Xu which is a component of u. From this it is essentially obvious that the L∞ (µ) norm on Xu agrees with its abstract M space norm. If we represent Xu as a C(K) space, then the sets of positive µ measure are mapped onto the nonempty clopen subsets of K in an obvious way so that we can represent X and L1 (µ) as function spaces on K and transfer the measure µ to K.

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