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In $\mathbb{R}^3$, can anyone help find a configuration of 5 lines such that the minimum of the smallest semi-axis lengths of the ellipsoid $ \mathbf{x}^T \mathbf{A} \mathbf{x} = 1 $, where $\mathbf{A}$ is the Gram of the matrix consisting of the unit directional vectors of any 3 lines, is maximized over all configurations of 5 lines?

Thank you for any helpful answers!

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    $\begingroup$ Isn't "the minimum of the smallest angles between any three lines" just the minimum angle between any two lines? $\endgroup$ Commented Sep 29, 2024 at 2:53
  • $\begingroup$ @RavenclawPrefect Thank you for your comment. It has been revised. Thanks! $\endgroup$ Commented Sep 29, 2024 at 3:33
  • $\begingroup$ So you are interested in unit directional vectors, not the lines themselves, don't you? $\endgroup$ Commented Oct 3, 2024 at 3:35
  • $\begingroup$ @MaxAlekseyev Yeah, but it seems to be more convenient to describe in lines. Thanks. $\endgroup$ Commented Oct 3, 2024 at 7:25

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