A simple geometric proof and some applications are given to results of C. Borell providing necessary and sufficient conditions that a density in Rⁿ generates a measure satisfying a convexity property of the type P(θ A₀ + (1 - θ)A₁) \θ P(A₀)ˢ + (1 - θ) P(A₁) ˢ\1/s.
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Yosef Rinott (1976) studied this question.