Generic Fréchet Differentiability of Convex Functions Dominated by a Lower Semicontinuous Convex Function

Generic G&eaux- Differentiability of Convex Functions on Small Sets

The theory of generic differentiability of convex functions on Banach spaces is by now a well-explored part of infinite-dimensional geometry. All the attempts to solve this kind of problem have in common, as a working hypothesis, one special feature of the finite-dimensional case. Namely, convex functions are always considered to be defined on convex sets with nonempty interior. But typically, ...

Generic Fr6chet -differentiability of Convex Functions on Small Sets

Lower Semicontinuous Convex Relaxation in Optimization

We relate the argmin sets of a given function, not necessarily convex or lower semicontinuous, and its lower semicontinuous convex hull by means of explicit characterizations involving an appropriate concept of asymptotic functions. This question is connected to the subdifferential calculus of the Legendre–Fenchel conjugate function. The final expressions, which also involve a useful extension ...

Differentiability of convex functions outside small sets

Small sets. We are going to present some results of the following type: under a suitable assumption on a normed space X, each continuous convex function, defined on an open convex set A ⊂ X, is (Gâteaux or Fréchet) differentiable outside a set which is in some sense small. It is natural to ask that a nonempty family S ⊂ 2X whose elements are considered “small sets” satisfy the following conditi...

A Convex, Lower Semicontinuous Approximation of Euler's Elastica Energy

We propose a convex, lower semi-continuous, coercive approximation of Euler’s elastica energy for images, which is thus very well-suited as a regularizer in image processing. The approximation is not quite the convex relaxation, and we discuss its close relation to the exact convex relaxation as well as the difficulties associated with computing the latter. Interestingly, the convex relaxation ...