2015/04/27 by Triloki Nath, Nath, Triloki
Computer Science · Mathematics · #41A65 #46B20 #FOS: Mathematics #Fixed Point Theorems Analysis #Functional Analysis (math.FA) #Functional Equations Stability Results #Optimization and Variational Analysis
paper · pdf · doi:10.48550/arxiv.1504.07292
openalex publication_date 2015/04/27 · openalex created_date 2022/10/02 · openalex updated_date 2026/07/28
We establish a necessary and sufficient condition for the differentiability\nof the distance function generated by a nonempty closed set K in a real normed\nlinear space X under a proximinality condition on K. We do not assume the\nuniform differentiability constraints on the norm of the space as in Giles\n[16]. Hence, our result advances that of Giles [16]. We prove that the\nproximinal condition of Giles [16] is true for almost suns. The proximinal\ncondition ensures convexity of an almost sun in some class of strongly smooth\nspaces under a differentiability condition of the distance function. A\nnecessary and sufficient condition is obtained for the convexity of Chebyshev\nsets in Banach spaces with rotund dual.\n