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arXiv · 2206.01596

On the value of the fifth maximal projection constant

Beata Derȩgowska·Matthew Fickus·Simon Foucart·Barbara Lewandowska
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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 $\lambda(5)$.

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Beata Derȩgowska, Matthew Fickus, Simon Foucart, Barbara Lewandowska. 2022-06-03. On the value of the fifth maximal projection constant. https://doi.org/10.1016/j.jfa.2022.109634

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