On the Proximinality of Ridge Functions
DOI:
https://doi.org/10.5644/SJM.05.1.10Keywords:
Ridge function, extremal element, proximinality, path, orbitAbstract
Using two results of Garkavi, Medvedev and Khavinson [7], we give sufficient conditions for proximinality of sums of two ridge functions with bounded and continuous summands in the spaces of bounded and continuous multivariate functions respectively. In the first case, we give an example which shows that the corresponding sufficient condition cannot be made weaker for some subsets of $\mathbb{R}^{n}$. In the second case, we obtain also a necessary condition for proximinality. All the results are illuminated by numerous examples. The results, examples and following discussions naturally lead us to a conjecture on the proximinality of the considered class of ridge functions.
2000 Mathematics Subject Classification. 41A30, 41A50, 41A63
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