Tikhonov Well-Posedness and Differentiability on Asymmetrically Normed Spaces

On normed vector spaces there is a well-known connection between the Tikhonov well-posedness of a minimisation problem and the differentiability of an associated convex conjugate function. We show how this duality naturally generalises to the setting of asymmetrically normed spaces and prove a universal differentiability property of the convex conjugate of the cumulant-generating function of a mean-zero measure on a locally convex space.