Abstract

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.

Author information

Contact details are reproduced from the original publication and may be historical.

Grigorii E. Ivanov

Dept. of Higher Mathematics, Moscow Institute of Physics and Technology, Institutski str. 9, Dolgoprudny – Moscow region, Russia 141700

givanov@mail.mipt.ru

Keywords

  • Weak convexity
  • Minkowski functional
  • infimal convolution
  • quasiball

Mathematics Subject Classification

41A50, 41A65, 52A21

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Suggested citation

G. E. Ivanov. “Weak Convexity of Sets and Functions in a Banach Space.” Journal of Convex Analysis 22 (2015), No. 2, 365–398. https://doi.org/10.68381/jca22017

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Published by Heldermann Verlag, 2015. Rights now held by Banach Press.