The reason I am confused is because in Thm-1.1, $\gamma_2$ is with respect to the natural pseudo-metric $d$ of the Gaussian Process, but for the rest of the paper ( for example Thm.-1.2) he considers $(\mathcal{X},\|.\|)$ to be a Banach space, and computes $\gamma_2$ bounds with respect to $d'(x,y) = \|x-y\|$. For the example in 3.1, he selected the Gaussian Process in such a way that $d$ coincides with the distance induced by $\|.\|_2$ in Euclidean space. So does this mean, these results are only applicable when $d$ is same as the metric induced by $\|.\|$ of some Banach space?

I have a question. Suppose $d$ is the natural metric of the Gaussian process, and $B(\rho) \subset \mathcal{X}$ is a $d$-ball, where $(\mathcal{X},\|.\|)$ is the Banach space indexing the Gaussian Process. Suppose the set $B(rho)$ is actually a symmetric convex subset of $\mathcal{X}$, and I can get good estimate of $\gamma_2$ with respect to the metric $\|x-y\|$ by Handel's paper. Can I use this to say anything about $\gamma_2$ with respect to the distance function $d$? (I am asking it because in one example I am trying out, $d$ balls correspond to ellipsoids in $\mathcal{X}$)

Thanks. I think I will first begin with simpler classes of covariance functions, for example covariance functions $k$ such that $k(x,y) = f(\|x-y\|)$ for some norm.