Weak Convexity of Sets and Functions in a Banach Space

We consider weakly convex sets with respect to (w.r.t.) a quasiball M (quasiball is a closed convex proper subset of a Banach space E with 0 being its interior point). We investigate the properties of M which are sufficient for equivalence of the weak convexity of a closed set A, single-valuedness and continuity of M-projection onto A from the M-tube around A, and Fréchet differentiability of the M-distance function on the M-tube around A. We show that a function f is weakly convex w.r.t. a convex function γ with γ(0)<0 iff the epigraph of f is weakly convex w.r.t. the epigraph of γ. The weak convexity of f w.r.t. a uniformly convex coercive function γ is characterized in terms of well posedness of the infimal convolution problem for f and γ.