Geodesic
Weak Convexity of Sets and Functions in a Banach Space
Journal of convex analysis, Tome 22 (2015) no. 2, pp. 365-398
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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.
Classification : 41A50, 41A65, 52A21
Mots-clés : Weak convexity, Minkowski functional, infimal convolution, quasiball 
@article{JCA_2015_22_2_JCA_2015_22_2_a2,
     author = {G. E. Ivanov},
     title = {Weak {Convexity} of {Sets} and {Functions} in a {Banach} {Space}},
     journal = {Journal of convex analysis},
     pages = {365--398},
     year = {2015},
     volume = {22},
     number = {2},
     url = {http://geodesic.mathdoc.fr/item/JCA_2015_22_2_JCA_2015_22_2_a2/}
}
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G. E. Ivanov. Weak Convexity of Sets and Functions in a Banach Space. Journal of convex analysis, Tome 22 (2015) no. 2, pp. 365-398. http://geodesic.mathdoc.fr/item/JCA_2015_22_2_JCA_2015_22_2_a2/