Solarity of Chebyshev Sets in Dual Spaces and Uniquely Remotal Sets
Alexey Rostislavovich Alimov · Lobachevskii Journal of Mathematics · 2021
Two max- and min-approximation problems on solarity of sets in dual spaces are considered. It is shown that if the metric projection onto a set $$M\subset X^{*}$$ is $$w^{*}$$ -upper semicontinuous and has nonempty $$w^{*}$$ -closed acyclic values, then $$M$$ is a sun. In particular, a Chebyshev set with $$w^{*}$$ -continuous metric projection is a sun. In the max-approximation setting, a set with $$w^{*}$$ -upper-semicontinuous $$\max$$ -projection with nonempty $$w^{*}$$ -closed acyclic values is shown to be local $$\max$$ -sun. As a result, it follows that that a uniquely remotal set with $$w^{*}$$ -continuous $$\max$$ -projection operator is a singleton, which gives an answer to the well-known unique farthest point problem in dual spaces for sets with $$w^{*}$$ -continuous farthest-point mapping.