On the proximinality of ridge functions

TL;DR

This paper investigates conditions under which sums of two ridge functions are proximinal in spaces of bounded and continuous functions, providing new sufficient and necessary conditions, examples, and a conjecture on the topic.

Contribution

It offers new sufficient and necessary conditions for the proximinality of sums of two ridge functions, along with illustrative examples and a conjecture.

Findings

Sufficient conditions for proximinality in bounded functions

Necessary conditions for proximinality in continuous functions

Counterexamples showing limitations of conditions

Abstract

Using two results of Garkavi, Medvedev and Khavinson, 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 furnished with plenty of examples. The results, examples and following discussions naturally lead us to a conjecture on the proximinality of the considered class of ridge functions. The main purpose of the paper is to draw readers' attention to this conjecture.