The Vector Measures Whose Range Is Strictly Convex

Let m be a measure on a measure space X, L.with values in Rn and f be the density of m with respect to its total variation. We show that the range R m.s m E.: E g L4 of m is strictly convex if and only if the determinant detwf x1., . . . , f xn.x is nonzero a.e. on X n. We apply the result to a class of measures containing those that are generated by Chebyshev systems.