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Iosif Pinelis
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More on the Gram matrix of $6$ unit vectors in $\Bbb R^3$

Let $G=(g_{ij}\colon i,j=1,\dots,6)$ be the $6\times6$ Gram matrix of $6$ unit vectors in $\Bbb R^3$. Let $$u:=\sum_{1\le i<j\le 6}g_{ij}^2,\quad v:=\sum_{1\le i<j<k\le 6}g_{ij}g_{ik}g_{jk}.$$

Can then $u-v$ be $<23/8$ given that $u\le15/4$?

Remark 1: A numerical experiment suggests that $23/8$ is the exact lower bound on $u-v$ given $u\le15/4$.

Remark 2: If the restriction "in $\Bbb R^3$" is removed, then, of course, one can make all the off-diagonal entries of $G$ zero.