Examples of Chebyshev Sets in Matrix Spaces

Abstract

Let A be a matrix in $\text{C}$^n×n and let UΣV* be its singular value decomposition. The authors prove that for each 1⩽k⩽n the set $\text{S}$^(k)₁={S∈$\text{C}$^n×n: ∑_{1⩽j1<…<jk⩽n} σ_j1(S) σ_j2(S)…σ_jk(S)⩽1} is a Chebyshev set in $\text{C}$^n×n endowed with the spectral norm and that the metric projection is globally Lipschitz-continuous.