# Minimax equivalence of Matrix Norm

In a proof of random matrix theory, the author makes use of the following equivalence:

$$$$\inf_{v\in V(r)}\lVert Xv \rVert_2 = \inf_{v\in V(r)} \sup_{u \in S^{n-1}}u^TXv$$$$ where $$X$$ is $$p X n$$ random matrix whose rows $$X_i$$ follows the distribution $$X_i \sim \mathcal{N}(0,K)$$ $$\forall i \in \{1,2,\ldots n\}$$ with $$S^{n-1} = \{ u \in \mathbb{R}^n | \lVert u \rVert = 1\}$$ and $$V(r) = \{ v \in \mathbb{R}^p | \lVert K^{1/2}v\rVert_2 = 1, \lVert v\rVert \leq r \}$$.

• Does this help? Also, the dual of the $\ell_2$ norm is the $\ell_2$ norm . – Vincent Guillemot May 21 '19 at 14:13
• There is a lot of irrelevant distracting material that hides a simpler statement: focus on vectors $v,$ matrices $X,$ and consider various equivalent definitions of the norm (as suggested by @Vincent). Ignore the random variables and the $\inf$ operations. – whuber May 21 '19 at 14:14