Hereditary properties of lower semicontinuous metric projection and solar properties of sets

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发表日期：

2025年1月8日

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A. R. Alimov

摘要

We how and when the lower semicontinuity of the metric projection operator is preserved upon intersections of a given set with balls, extreme planes, and, in the most general case, with bars (a bar is the intersection of some families of hyperstrips generated by extreme functionals of the dual unit sphere; a particular case of a bar is a closed ball). The results on hereditary properties of lower semicontinuous metric projection are obtained in the space $\ell ^\infty _n$ $ℓ_{n}^{\infty}$ , two-dimensional normed spaces, and three-dimensional space with cylindrical norm. We also construct an example of a normed space of any finite dimension $\ge 3$ $\geq 3$ in which lower semicontinuity of the metric projection is not preserved upon intersections with closed balls.