On the Differentiability of Saddle and Biconvex Functions and Operators

We strengthen and generalize results of J. M. Borwein [ Generic differentiability of order-bounded convex operators, J. Austral. Math. Soc. Ser. B 28 (1986) 22–29] and of A. Ioffe and R. E. Lucchetti [Typical convex program is very well posed, Math. Program. 104 (2005) 483–499] on Fréchet and Gâteaux differentiability of saddle and biconvex functions (and operators). For example, we prove that in many cases (also in some cases which were not considered before) these functions (and operators) are Fréchet differentiable except for a Γ-null, σ-lower porous set. Moreover, we prove these results for more general "partially convex (up or down)" functions and operators defined on the product of n Banach spaces.