Hi,

I’ve encountered the following assumption:

Let D be a set such that there exists a Minkowski function $f(u)$ on $\mathbb{R}^l$ and norm $g(v)$ on $\mathbb{R}^m$ such that $\forall u\in \mathbb{R}^l, \forall v\in \mathbb{R}^m$ $ \max_{X\in D} u^{t}Xv = f(u)g(v)$.

According to the author, there are many functions which accomplish this assumption.

One example is the set $D=\left( X| \Vert{X\Vert}_F \leq 1 \right)$ for which $f(u)=\Vert u \Vert_2$, $g(v)=\Vert v\Vert_2$. The upper bound $\max_{X\in D} u^{t}Xv \leq f(u)g(v)$ is easy to show by applying the Cauchy Schwartz inequality and then the consistency of the Frobenius norm. However, I can’t find an example to show a member in the set D attains an equality. That is, that $ \exists X\in D ,\ u^{t}Xv = f(u)g(v)$.

Can anyone help prove this assumption for this set (or any other)?

Thanks