  Let $\lambda_\mathbb{K}(m)$ denote the maximal absolute projection constant
over the subspaces of dimension $m$. Apart from the trivial case for $ m=1$,
the only known value of $\lambda_\mathbb{K}(m)$ is for $ m=2$ and
$\mathbb{K}=\mathbb{R}.$ In 1960, B.Grunbaum conjectured that
$\lambda_\mathbb{R}(2)=\frac{4}{3}$ and in 2010, B. Chalmers and G. Lewicki
proved it. In 2019, G. Basso delivered the alternative proof of this
conjecture. Both proofs are quite complicated, and there was a strong belief
that providing an exact value for $\lambda_\mathbb{K}(m)$ in other cases will
be a tough task. In our paper, we present an upper bound of the value
$\lambda_\mathbb{K}(m)$, which becomes an exact value for the numerous cases.
The crucial will be combining some results from the articles [B. Bukh, C. Cox,
Nearly orthogonal vectors and small antipodal spherical codes, Isr. J. Math.
238, 359-388 (2020)] and [G. Basso, Computation of maximal projection
constants, J. Funct. Anal. 277/10 (2019), 3560-3585.], for which simplified
proofs will be given.
