On paraconvexity in spaces of summable mappings

A new selection theorem is proved for lower semicontinuous mappings F:X→LM(T;B) into an Orlisz spaces of summable mappings. Values F(x) of F in this theorem in general are neither convex nor decomposable, they are unions of two sets which are both convex and decomposable. The key ingredient of the proof is an appropriate estimate of nonconvexity of such union.