On continuous choice of retractions onto nonconvex subsets

AbstractFor a Banach space B and for a class A of its bounded closed retracts, endowed with the Hausdorff metric, we prove that retractions on elements A∈A can be chosen to depend continuously on A, whenever nonconvexity of each A∈A is less than 12. The key geometric argument is that the set of all uniform retractions onto an α-paraconvex set (in the spirit of E. Michael) is α1−α-paraconvex subset in the space of continuous mappings of B into itself. For a Hilbert space H the estimate α1−α can be improved to α(1+α2)1−α2 and the constant 12 can be replaced by the root of the equation α+α2+α3=1