On continuous choice of retractions onto nonconvex subsets

For 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 1 2 . The key geometric argument is that the set of all uniform retractions onto an α-paraconvex set (in the spirit of E. Michael) is α 1 − α -paraconvex subset in the space of continuous mappings of B into itself. For a Hilbert space H the estimate α 1 − α can be improved to α ( 1 + α 2 ) 1 − α 2 and the constant 1 2 can be replaced by the root of the equation α + α 2 + α 3 = 1 .