Renorming, Proximinal Subspaces, and Quotients in Banach Spaces

Any Banach space can be equivalently renormed so that a given subspace is proximinal. We give a sufficient condition for an equivalent norm on 11(Γ) to satisfy the following property: every Banach space with the density character not greater than card (Γ) is isometric to a quotient of the renormed 11(Γ). Applications concerning the structure of quotients with proximinal subspaces are given