On the value of the fifth maximal projection constant

TL;DR

This paper establishes a new lower bound for the fifth maximal projection constant using a novel construction of equiangular tight frames, supporting the conjecture that this bound is exact.

Contribution

Introduces a new construction of mutually unbiased equiangular tight frames to determine a lower bound for ive, advancing understanding of projection constants.

Findings

ive bgeq 2.06919

The new bound matches numerical estimates, suggesting it is exact

Supports the conjecture about ive's precise value

Abstract

Let $\lambda(m)$ denote the maximal absolute projection constant over real $m$-dimensional subspaces. This quantity is extremely hard to determine exactly, as testified by the fact that the only known value of $\lambda(m)$ for $m>1$ is $\lambda(2)=4/3$. There is also numerical evidence indicating that $\lambda(3)=(1+\sqrt{5})/2$. In this paper, relying on a new construction of certain mutually unbiased equiangular tight frames, we show that $\lambda(5)\geq 5(11+6\sqrt{5})/59 \approx 2.06919$. This value coincides with the numerical estimation of $\lambda(5)$ obtained by B. L. Chalmers, thus reinforcing the belief that this is the exact value of $\lambda(5)$.