Given nonsingular $A\in\mathbb{R}^{d\times d}$, I would like to compute the optimal value of the following $$\underset{\|y\|_2 = 1}{\max} \|Ay\|_{p} - \|Ay\|_{q}, \quad 1 \leq p < q < \infty,$$ as function of the matrix $A$. Instead of optimizing over unit vectors, one can optimize over the ball $\|y\|_2 \leq 1$ without loss of generality because the objective is positive homogeneous in $y$.
If the matrix $A$ were identity, then we can appeal to the norm inequality $\|\cdot\|_{p} \leq d^{1/p - 1/q}\|\cdot\|_{q}$, $1 \leq p < q < \infty$, which is a consequence of the Hölder's inequality. Since the upper bound in this inequality in $\mathbb{R}^{d}$ is achieved by any point $v\in\{-1,1\}^{d}$, so we expect the maximizing unit vector in this case to be $y_{\max} = \alpha v$ for any $v\in\{-1,1\}^{d}$, where the normalizing constant $\alpha=1/\sqrt{d}$. So the optimal value in this special case should be $(d^{1/p} - d^{1/q})/\sqrt{d}$, which matches with numerics.
However, I am not sure how to reason in the case of generic nonsingular $A$. The heuristic $y_{\max} = \alpha A^{-1}v$ in this case leads to the value $\dfrac{d^{1/p} - d^{1/q}}{\sqrt{\underset{v\in\{-1,1\}^{d}}{\min}v^{\top}\left(AA^{\top}\right)^{-1}v}}$, which does not match with the numerics.
Any ideas or references are welcome. I have looked into "Differences of means" by Shisha and Mond, Bulletin of the AMS, 1967, but the setting there seemed quite different and not directly relevant to our case.