# volumes of balls under an affine transformation

Denote by $B_t(O,\rho) \subset \mathbb{R}^t$, the sphere centered at the origin with radius $\rho$, and $B_n(O,\delta) \subset \mathbb{R}^n$, the sphere centered at the origin with radius $\delta$.

For a $b \in \mathbb{R}^n$ and $M$ a $n \times t$ matrix, whose $d$-th largest singular value is $\sigma$, define the map $A:\mathbb{R^t}\rightarrow\mathbb{R}^n$ by $$F(x)=Mx + b.$$

Is there a tight bound for the volume of the intersection of $B_t(O,\rho)$ and the pre-image of $B_n(O,\delta)$ under $F$?

My difficulty here (I think) is I interpret $B_t(O,\rho) \cap F^{-1}(B_n(O,\delta))$ as vectors in the $\rho$-ball who are mapped to the $\delta$-ball in $\mathbb{R}^n$. In particular, $M$ sends the $\rho$-ball to a $d$-dimensional ellipsoid in $\mathbb{R}^t$ which is then translated by $b$. Is there a better way of interpeting this set which can help me get the bound that I need?

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