The Vector Measures whose Range is strictly convex

Let μ be a measure on a measure space (X,Λ) with values in Rnandfbe the density of μ with respect to its total variation. We show that the range R(μ)={μ(E):E∈Λ} of μ is strictly convex if and only if the determinant det[f(x1),...,f(xn)] is nonzero a.e. onXn. We apply the result to a class of measures containing those that are generated by Chebyshev systems.