Geodesic
A separation theorem for nonconvex sets and its applications
Fundamentalʹnaâ i prikladnaâ matematika, Tome 21 (2016) no. 4, pp. 23-66

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We prove theorems on separation by sphere or (in a general case) by the boundary of a shifted quasiball of two closed disjoint subsets of a Banach space one of which is prox-regular or weakly convex and the other is a summand of a ball or quasiball. These separation theorems are applied for proving some theorems on the continuity of the intersection of two multifunctions, the values of one of them being prox-regular or weakly convex (nonconvex, in general), and the values of the other being convex and summands of a ball or quasiball. As a corollary, a theorem on the continuity of a multifunction with values bounded by the graphs of two functions is obtained. 
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     author = {G. E. Ivanov and M. S. Lopushanski},
     title = {A separation theorem for nonconvex sets and its applications},
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G. E. Ivanov; M. S. Lopushanski. A separation theorem for nonconvex sets and its applications. Fundamentalʹnaâ i prikladnaâ matematika, Tome 21 (2016) no. 4, pp. 23-66. http://geodesic.mathdoc.fr/item/FPM_2016_21_4_a2/