Computation of maximal projection constants Giuliano Basso January 24, 2019 Abstract The linear projection constant Π(E) of a finite-dimensional real Ba- nach 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 Π n the maximal linear projection constant amongst n-dimensional Banach spaces. In this article, we prove that Π n may be determined by computing eigenvalues of certain two-graphs. From this result we obtain that the relative projection constants of codi- mension n converge to 1 + Π n . Furthermore, using the classification of K 4 -free two-graphs, we give an alternative proof of Π 2 = 4 3 . We also show by means of elementary functional analysis that for each integer n  1 there exists a polyhedral n-dimensional Banach space F n such that Π(F n )= Π n . 1 Introduction 1.1 Overview As a consequence of ideas developed by Lindenstrauss, cf. [Lin64], for a finite-dimensional Banach space E ⊂ ℓ ∞ (N) the smallest con- stant C ∈ [0, +∞) such that E is an absolute C-Lipschitz retract is completely determined by the linear theory of E. Indeed, Rieffel, cf. [Rie06], established that it is equal to the linear projection constant of E, which is the number Π(E) ∈ [0, +∞] defined as inf  ‖P‖ | P : ℓ ∞ (N) → E bounded surjective linear map with P 2 = P  . Linear projections have been the object of study of many researchers and the literature can be traced back to the classical book by Banach, cf. [Ban32, p.244-245]. The question about the maximal value Π n of the linear projection constants of n-dimensional Banach spaces has persisted and is a notoriously difficult one. In this article, we establish a formula that relates Π n with eigenvalues of certain two-graphs. This reduces the problem (in principle) to the classification of certain two-graphs and thus allows the 1 arXiv:1901.07866v1 [math.MG] 23 Jan 2019