Uniformly convex sets in Banach spaces

A formula is obtained expressing the modulus of smoothness of the support function of a set using the modulus of convexity of this set. This formula generalizes the well-known formula of J. Lindenstrauss that expresses the modulus of smoothness of the dual space using the modulus of convexity of the original space. An exact relationship is obtained among the exponent of uniform convexity of a set, the Hölder exponent for the derivative of its support function, the exponent of smoothness of this support function, and other exponents characterizing this set.