This is an announcement for the paper "Differentiability of Distance Function and The Proximinal Condition implying Convexity" by Triloki Nath.
Abstract: A necessary and sufficient condition for the differentiability of the distance function generated by an almost proximinal closed set has been given for normed linear spaces with locally uniformly convex and differentiable norm. We prove that the proximinal condition of Giles [6] is true for almost sun. In such spaces if the proximinal condition is satisfied and the distance function is uniformly differentiable on a dense set then it turn in the differentiability on all off the set (generating the distance function). The proximinal condition ensures about the convexity of almost sun in some spaces under a differentiability condition of the distance function. A necessary and sufficient condition is derived for the convexity of Chebyshev sets in Banach spaces with rotund dual.
Archive classification: math.FA
Mathematics Subject Classification: 41A65, 46B20
Remarks: 9 pages
Submitted from: tnath@dhsgsu.ac.in
The paper may be downloaded from the archive by web browser from URL
http://front.math.ucdavis.edu/1504.07292
or