Banach November 2008

banach@mathdept.okstate.edu

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Abstract of a paper by Dusan Repovs and Pavel V. Semenov
by alspach＠fourier.math.okstate.edu 04 Nov '08

04 Nov '08

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(a)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(a)arXiv.org.

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Banach November 2008