Let $f:\mathbb{R}^D\rightarrow\mathbb{R}$ be strongly convex with respect to a (not necessarily Euclidean) norm $\|\cdot\|$, so that $$ f(y) \geq f(x)+\nabla f(x)^T(y-x)+\frac{\mu}{2} \|y-x\|^2 $$ is satisfied for all $x,y$. For the Euclidean norm, one could also write the equivalent condition $$ (\nabla f(x)-\nabla f(y))^T(x-y) \geq \mu\|x-y\|^2. $$ Does a similar condition hold for strong convexity with respect to an arbitrary $\ell_p$ norm?
More specifically, I'm trying to prove that the inverse operator $(\nabla f)^{-1}$ is a function and is Lipschitz with $\frac{1}{\mu}$, as in the case with the Euclidean norm. Is a similar result possible in the general case?