Journal of Convex Analysis 22 (2015), No. 2, 365--398
Copyright Heldermann Verlag 2015
Weak Convexity of Sets and Functions in a Banach Space
Grigorii E. Ivanov
Dept. of Higher Mathematics, Moscow Institute of Physics and Technology, Institutski str. 9, Dolgoprudny -- Moscow region, Russia 141700
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.
Keywords: Weak convexity, Minkowski functional, infimal convolution, quasiball.
MSC: 41A50, 41A65, 52A21
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