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We describe the class of all functions φ: [0,∞) → R for which the second derivative g′′ φ (x; y, y) of the spherically symmetric function gφ (x) := φ(|x|) in the direction of y is convex in x , where x and y are vectors in a Hilbert space H and | · | is the norm in H . Applications to extremal probabilistic problems are given. Mathematics subject classification (2000): Primary: 26B25, 60E15; Secondary: 26D10, 34A40, 52A40, 52A41.

Spherically symmetric functions with a convex second derivative and applications to extremal probabilistic problems