Abstract
We characterize the subsets of the space \(\ell^\infty_n\) with continuous (lower semicontinuous) metric projection. One of the characteristic properties is the strict solarity of both the set and any nonempty intersection thereof with any support coordinate hyperplane.
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Notes
The acronym “RIF” stands for “relative interior of a face.”
The acronym “SHSS” means “support hyperplane” and “strict sun.”
The definition of a cocross is given below.
A set is called a \(B\)-sun if every nonempty intersection of \(M\) with any closed ball is a sun.
The linear dimension of a convex set \(C\subset\ell^\infty_n\) is the dimension of a maximal affine subspace contained in \(C\). For a support cone \(\mathring{K}(x,y)\) in \(\ell^\infty_n\), such an affine subspace can always be chosen in the set of coordinate subspaces.
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Acknowledgments
The author is grateful to I. G. Tsar’kov for useful discussions.
Funding
This work was supported by the Russian Foundation for Basic Research under grants 19-01-00332-a and 18-01-00333-a.
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Alimov, A.R. Characterization of Sets with Continuous Metric Projection in the Space \(\ell^\infty_n\). Math Notes 108, 309–317 (2020). https://doi.org/10.1134/S0001434620090011
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DOI: https://doi.org/10.1134/S0001434620090011