# Computation of maximal projection constants

## TL;DR

This paper investigates the maximal projection constants of finite-dimensional Banach spaces, linking them to eigenvalues of two-graphs, and provides new methods to compute and understand these constants.

## Contribution

It introduces a novel approach to determine maximal projection constants using eigenvalues of two-graphs and offers an alternative proof for a specific case, expanding the theoretical understanding.

## Key findings

- Maximal projection constants relate to eigenvalues of two-graphs.
- Convergence of relative projection constants to 1+Pi_n.
- Existence of polyhedral spaces achieving 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 $n\geq 1$ there exists a polyhedral $n$-dimensional Banach space $F_n$ such that $\Pi(F_n)=\Pi_n$.