dual relationship $\|x\|_\infty = \max_{\|z\|_1 \le 1} z^T x$

Question

I was reading the answer in the stackexchange question about dual cones of L-1 norm cone

It says the key is the dual relationship $\|x\|_\infty = \max_{\|z\|_1 \le 1} z^T x$.

I am trying to understand what it means but with zero success. For simplicity I am assuming a 2D vector space (R2).

$\textbf{My attempt in understanding}$:

lhs = The largest entry in the vector (x1,x2)

rhs= maximum value of dot product for any vector x with another vector z taken from a set of vectors with L1 norm less than or equal to unity.

Am I correct in my understanding? I am not able to understand why this relation is true. Could someone explain this?

Answer 1

We have

$$z^Tx= \sum_{i=1}^n z_ix_i\le \sum_{i=1}^n |z_i||x_i| \le \|x\|_\infty\sum_{i=1}^n |z_i|=\|x\|_\infty \|z\|_1\le \|x\|_\infty$$

Now, we just need to exhibit a particular $z$ to make that inequality holds.

If $x = 0$, the case is obvious.

Suppose not, we can just find a particular index $j$ such that $x_j$ that satisfies $|x_j|= \|x\|_\infty$ and construct $z$ such that $z_j = sign(x_j)$ and $z_i = 0$ if $i \ne j$, then $$z^Tx = z_j x_j =sign(x_j)x_j = |x_j|=\|x\|_\infty.$$

Remark:

note that LHS correspondd to the largest magnitude.