For what $p$ is the condition number of a given matrix $A$, using the $p$ norm on matrices, minimal?

Question

For what p is the condition number of a given matrix $A$, using the p norm on matrices, minimal?

The condition number on $A$ is given as:

$$K(A,p) = \|A\|_p \times \|A^{-1}\|_p$$

I tried simplifying this further but got nowhere, I feel like I'm missing something. Any help would be appreciated!