A function $f$ is said to be strongly convex with respect to a norm $\|\cdot\|_p$ if for all $x,y$, $$f(x) \geq f(y) + \nabla f(y)^T(x-y) + \frac{1}{2}\|x-y\|^2_p.$$

There are a bunch of functions used in machine learning, statistics, etc. that are extremely well known to be strongly convex with respect to the $2$ or $1$ norm

Examples:

$\sum_{j = 1}^m x_j^2$ is 2-strongly convex with respect to $\|\cdot\|_2$

$\sum_{j = 1}^m x_j \log(x_j) $ is 1-strongly convex with respect to $\|\cdot\|_1$

Does there exist any strongly convex function with respect to a norm $\|\cdot\|_p, p>2$?