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It is known that, in finite dimensions, the support function of a compactconvex set with non empty interior is differentiable excepting the origin ifand only if the set is strictly convex. In this paper we realize a thoroughstudy of the relations between the differentiability of the support function onthe interior of its domain and the convexity of the set, mainly for unboundedsets. Then we revisit some results related to the differentiability of the costfunction associated to a production function.

Relations between the convexity of a set and the differentiability of its support function