Let $\Sigma \in \mathbb{R}^{n \times n}$ be a symmetric, positive definite matrix and let $\mu_r$ denote surface measure on the sphere in $\mathbb{R}^n$ with radius $r$. Let $$ R = \{x \in \mathbb{R}^n : |x_i| \leq a_i, \ i = 1, \dots, n\} $$ denote an $n$-dimensional rectangle that centers the origin.

Many geometric properties are known when considering $\Sigma$ as a linear map from $\mathbb{R}^n$ to itself, in particular the map will transform a sphere to an ellipse, rotate the standard basis and scale them proportional to the eigenvalues, etc. As such there are many reformulations of my question, which is the following:

Ideally, I would like $\mu_r(\Sigma^{-1}R)$ as explicitly as possible given in terms of $r$ and the elements of $\Sigma$ and the $a_i.$

Geometrically, this is the surface area of the points on the sphere with corresponding points on the ellipsoid (under $\Sigma$) contained in $R.$

Other comments or answers to related problems are welcome!


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