Inversion formulae for Siegel transforms

Let $n\in {\mathbb{Z}}_{\ge 3}$ be given. We prove Lebesgue-almost everywhere pointwise inversion formulae for the Siegel transforms in the geometry of numbers. These inversion formulae are quite general; for instance, they are valid for the Siegel transforms of any even and compactly supported Borel measurable function $f:{\mathbb{R}}^n\to \mathbb{R}$ that belongs to $L^1({\mathbb{R}}^n)\cap L^2({\mathbb{R}}^n)\mathrm{.}$