I am embarassed to pose this question. It is a generalization of a question asked less than 24 hours ago by an *unknonw (Google)*, which has been deleted since then, presumably by its author (her/)himself.

Let $M\in M_n(\mathbb R)$ be given. A version of the question can be written for non-square matrices. Complex entries may be considered as well. Its singular values $s_1\ge s_2\ge \cdots\ge s_n(\ge0)$ are given by the Rayleigh--Weyl formula $$s_k=\inf_{\dim F=n+1-k}\sup_{\quad x\in F|x\ne0} \frac{\|Mx\|_2}{\|x\|_2}=\sup_{\dim F=k}\inf_{\quad x\in F|x\ne0} \frac{\|Mx\|_2}{\|x\|_2},$$

where $\|\cdot\|_p$ stands for the $\ell^p$-norm over $\mathbb R$.

When $p\in[1,+\infty]$, $\ell^p$ version of the singular values, denoted $s_{k,p}(M)$, could be defined the same way, but replacing the $\ell^2$-norm by the $\ell^p$ one. When $p=2$, we know $s_kM)=s_k(M^T)$. Hence the question:

Is there a relation between $s_{k,p}(M)$ and $s_{k,p'}(M^T)$, when $p$ and $p'$ are conjugate exponent ?

A first attempt, unsuccessful, is to pretend that given $F$ or dimension $k$, there exists a $G$ of the same dimension such that for every $x\in F$, one has $$\|x\|_{p}=\sup_{y\in G|y\neq 0}\frac{y^Tx}{\|y\|_{p'}}.$$ Unfortunately, this is false in most cases, even thoug it is true for $p=2$ (take $G=F$) and for $k=1$ (Hahn-Banach).