This is an announcement for the paper "On continuous choice of retractions onto nonconvex subsets" by Dusan Repovs and Pavel V. Semenov. Abstract: 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 $A \in \A$ can be chosen to depend continuously on $A$, 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 $B$ into itself. For a Hilbert space $H$ 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 $\a+ \a^{2}+a^{3}=1$. Archive classification: math.GN math.FA Mathematics Subject Classification: 54C60; 54C65; 41A65; 54C55; 54C20 The source file(s), VerzijaZaArhiv.tex: 38914 bytes, is(are) stored in gzipped form as 0810.3895.gz with size 12kb. The corresponding postcript file has gzipped size 89kb. Submitted from: dusan.repovs@guest.arnes.si The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/0810.3895 or http://arXiv.org/abs/0810.3895 or by email in unzipped form by transmitting an empty message with subject line uget 0810.3895 or in gzipped form by using subject line get 0810.3895 to: math@arXiv.org.