Continuity and Selections of the Intersection Operator Applied to Nonconvex Sets

For a convex body

in a Banach space

E E

we consider the class

{\mathcal{S}}(C) S ( C )

of closed sets

A\subset E A ⊂ E

satisfying the support condition with respect to

C C

. If

C C

is a ball with radius

r r

, then

{\mathcal{S}}(C) S ( C )

is exactly the class of uniformly

r r

-prox-regular sets. We prove that the intersection operator

(A,C)\mapsto A\cap C ( A , C ) ↦ A ∩ C

is uniformly Hausdorff continuous and has a uniformly continuous selection on the family of pairs

(A,C) ( A , C )

such that

C C

is closed and uniformly convex,

rA\in{{\mathcal{S}}}(C) r A ∈ S ( C )

with

r\in(0,1) r ∈ ( 0 , 1 )

, and

A\cap C\ne\emptyset A ∩ C ≠ ∅

. We also deduce some new sufficient condition for affirmative solution of the splitting problem for selections.