Let be $F$ a compact set of $\Bbb R^n$ and $\mu$ the $(n - 1)$-dimensional Hausdorff measure.
If I denote $P_V : \Bbb R^n \to \Bbb R^n$ the orthogonal projection on $V$ for $V \subset \Bbb R^n$ a subspace of dimension $m \leq n$.
I'd like to know if there is a way to describe roughly what would look $\mu \circ P_{\mathcal{H}}$ where $\mathcal{H}$ is a hyperplane of normal $\vec{n}$.
In other words, let be $\Phi_F : \vec{n} \mapsto \mu\left(P_{\vec{n}^{\perp}}(F)\right)$, I'd like to describe this map.
I tried to see simple cases ($n = 2$), and to try to take approximations/cuts of $F$ to approximate the result, but didn't succeed.
I'm not sure there is even an proper answer to this question.