Projection Constants and Banach-Mazur Distance in Codimension Two

Abstract

We prove that every subspace $V\subset {\ell }_{\infty }^N$ of codimension one or two with absolute projection constant $\lambda (V)<4/3$ satisfies $$d(V,{\ell }_{\infty }^{\dim V})\le \frac{\lambda (V)}{4-3\lambda (V)}.$$ We conjecture that the codimension assumption can be removed and that the same estimate holds for every subspace $V\subset {\ell }_{\infty }^N$ with $\lambda (V)<4/3$.