I'm struggling to understand how can I compute the instrinsic volumes (or Minkowski functionals) of a submanifold $\Omega$ of $\mathbb{R}^N$. I found a formula, but I really can't understand it...
It is like:
$$M_n(\Omega)=\dfrac{N!\omega_N}{(N-n)!\omega_{N-n}}\int_{R^N} \phi(\Omega\cap gE_n)\mu_N(dg),$$
where $\phi$ is the Euler Characteristic, $\omega_n$ the volume of the n-dimensional unit ball, $E_n$ is any n-dimensional affine subspace of $R^N$, and $\mu_N$ is the Haar measure on $\mathbb{R}^N$. Nothing about $g$ is specified in my source.
It is above my level to compute an intrinsic volume via this formula...
I know that the volume and the surface area are intrinsic volumes. I want something simplified, using integration w.r.t Lebesgue measure, or Hausdorff measure and not in terms of Euler Characteristic which is hard to handle, like:
$J(\Omega)=\displaystyle\int_{\Omega} 1 dx$ (the n-dimensional volume)
$J(\Omega)=\displaystyle\int_{\partial\Omega} 1 d\sigma$ (the area)
Can be the above formula simplified for sets with a good structure?