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$\newcommand\R{\mathbb R}\newcommand\Q{\mathcal Q}$For mutually orthogonal vectors unit vectors $a=[a_1,\dots,a_n]^T$ and $b=[b_1,\dots,b_n]^T$ in $\R^n=\R^{n\times1}$ (so that $n\ge2$) and for all $x=[x_1,x_2]^T\in\R^2$, let $$\|x\|_{a,b}:=\sum_{i=1}^n|a_i x_1+b_i x_2|\quad\text{and}\quad \|x\|_1:=|x_1|+|x_2|.$$ Let $$N_{a,b}:=\max\{\|x\|_1\colon x\in\R^2,\,\|x\|_{a,b}\le1\},$$ the norm of the identity operator on $\R^2$ with respect to the norms $\|\cdot\|_{a,b}$ and $\|\cdot\|_1$.

It was shown that the exact upper bound on $N_{a,b}$ over all mutually orthogonal unit vectors $a$ and $b$ in $\R^n$ is $\sqrt2$.

Now, for mutually orthogonal unit vectors $c$ and $d$ in $\R^n$, let $\Q_{c,d}$ denote the set of all orthogonal matrices $Q\in\R^{n\times n}$ whose first two columns are $c$ and $d$. For any $Q\in\Q_{c,d}$, let $a_Q$ and $b_Q$ denote the first two columns of the matrix $Q^T$.

Question: Can one give an explicit expression for $\min\{ N_{a_Q,b_Q}\colon Q\in\Q_{c,d}\}$?

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  • $\begingroup$ To me it seems more natural to denote $V = [a_i\ b_i]\in\mathbb{R}^{n\times 2}$, assume $V^TV = I_2$ and write $\norm{x}_{a,b} = \|Vx\|_1$. $\endgroup$
    – Dirk
    Dec 14, 2020 at 15:55
  • $\begingroup$ @Dirk : I am not sure what your notation $V=[a_i,b_i]$ means, but I guess, together with the condition $V^TV=I$, it was supposed to mean just that $a$ and $b$ are mutually orthogonal unit vectors in $\mathbb R^n$. I am not sure why the $V$ notation would be better than the $a,b$ notation. Anyhow, the question is about, not $N_{a,b}$ per se, but $N_{a_Q,b_Q}$ for $Q\in\mathcal Q_{c,d}$. $\endgroup$
    – Iosif Pinelis
    Dec 14, 2020 at 19:39
  • $\begingroup$ I agree, this was mainly a note for myself, and I thought, that others might find it helpful, too. $\endgroup$
    – Dirk
    Dec 14, 2020 at 20:37

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