Let $A$ and $B$ in $\mathbb{R}^{d\times d}$ be positive semi-definite (psd) matrices and let $d\tau$ be the uniform probability distribution on the unit sphere $\mathbb{S}^{d-1}$ in $\mathbb{R}^d$. I am interested in the quantity $$ S(A,B)^2 := \int_{\mathbb{S}^{d-1}} (\Vert \theta\Vert_A - \Vert \theta\Vert_B)^2 d\tau(\theta) $$ where $\Vert \theta \Vert_A = \sqrt{\theta^\top A\theta}$ is the Mahalanobis (semi)-norm on $\mathbb{R}^d$. Is it possible to give a simpler expression for this quantity? Or at least non-trivial lower bounds?
Where this comes from: A well known distance between two psd matrices $A$ and $B$ is the Bures-Wasserstein distance given by $$ D(A,B)^2 := \mathop{tr}{A} + \mathop{tr}{B} - 2 \mathop{tr}{(A^{1/2}BA^{1/2})^{1/2}}. $$ It equals to the $2$-Wasserstein distance between the Gaussian distributions $\mathcal{N}(0,A)$ and $\mathcal{N}(0,B)$ (see, e.g.~Bhatia et al.). Here, I am interested by the "sliced" or "Radon transform" version of this distance, obtained by averaging the squared-distance over all $1$ dimensional projections of the Gaussians (which, for a projection direction $\theta \in \mathbb{S}^{d-1}$, are Gaussians of variance $\theta^\top A\theta$ and $\theta^\top B\theta$). The expression above indeed satisfies $$ S(A,B)^2 = \int_{\mathbb{S}^{d-1}}D(\theta^\top A \theta, \theta^\top B \theta)^2 d\tau(\theta). $$
