On continuous choice of retractions onto nonconvex subsets
arXiv:0810.3895 · doi:10.1016/j.topol.2009.04.066
Abstract
For a Banach space and for a class $\A$ of its bounded closed retracts, endowed with the Hausdorff metric, we prove that retractions on elements $A \in \A$ can be chosen to depend continuously on , whenever nonconvexity of each $A \in \A$ is less than $\f{1}{2}$. The key geometric argument is that the set of all uniform retractions onto an $\a-$paraconvex set (in the spirit of E. Michael) is $\frac{\a}{1-\a}-$paraconvex subset in the space of continuous mappings of into itself. For a Hilbert space the estimate $\frac{\a}{1-\a}$ can be improved to $\frac{\a (1+\a^{2})}{1-\a^{2}}$ and the constant $\f{1}{2}$ can be reduced to the root of the equation $\a+ \a^{2}+a^{3}=1$.