Properties of best approximations with respect to the Ky Fan $p$-$k$ norm, and the strict spectral approximant of a matrix

Some questions raised in [K. Ziętak, {\it From the strict Chebyshev approximant of a vector to the strict spectral approximant of a matrix}, Warsaw : Banach Center Publ., 112 Polish Acad. Sci. Inst. Math. (2017)] are discussed. To do so, the subdifferential set of the Ky Fan $p$-$k$ norm is computed. A characterization for the best approximations with respect to the Ky Fan $p$-$k$ norms is given. Further, necessary and sufficient conditions for $\epsilon$-Birkhoff orthogonality with respect to the Ky Fan $p$-$k$ norm are also derived.