Projection Constants and Banach-Mazur Distance in Codimension Two

Abstract: We prove that every subspace $V\subset\ell_\infty<sup>N$ of codimension one or two with absolute projection constant $λ(V)&lt;4/3$ satisfies $$d(V,\ell_\infty^{\dim V}) \leq\frac{λ(V)}{4-3λ(V)}.$$ We conjecture that the codimension assumption can be removed and that the same estimate holds for every subspace $V\subset\ell_\infty<sup>N$ with $λ(V)&lt;4/3$.