Computation of maximal projection constants

Abstract

The linear projection constant Π(E) of a finite-dimensional real Banach space E is the smallest number C∈[0,+∞) such that E is a C-absolute retract in the category of real Banach spaces with bounded linear maps. We denote by Πn the maximal linear projection constant amongst n-dimensional Banach spaces. In this article, we prove that Πn may be determined by computing eigenvalues of certain two-graphs. From this result we obtain that the maximal relative projection constants of codimension n converge to 1+Πn. Furthermore, using the classification of K4-free two-graphs, we give an alternative proof of Π2=43. We also show by means of elementary functional analysis that for each integer n⩾1 there exists a polyhedral n-dimensional Banach space Fn such that Π(Fn)=Πn.