Computation of maximal projection constants

Abstract

The linear projection constant $\Pi (E)$ of a finite-dimensional real Banach space $E$ is the smallest number $C\in [0,+\infty )$ such that $E$ is a $C$-absolute retract in the category of real Banach spaces with bounded linear maps. We denote by ${\Pi }_n$ the maximal linear projection constant amongst $n$-dimensional Banach spaces. In this article, we prove that ${\Pi }_n$ may be determined by computing eigenvalues of certain two-graphs. From this result we obtain that the relative projection constants of codimension $n$ converge to $1+{\Pi }_n$. Furthermore, using the classification of $K_4$-free two-graphs, we give an alternative proof of ${\Pi }_2=\frac{4}{3}$. We also show by means of elementary functional analysis that for each integer $n\ge 1$ there exists a polyhedral $n$-dimensional Banach space $F_n$ such that $\Pi (F_n)={\Pi }_n$.

27 pages

Cited by in corpus (8)

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• Minimal projections onto spaces of polynomials on real euclidean spheres
• Erratum to "Computation of maximal projection constants"