I'm reading a book, and the author consider the following situation: Let $K$ be a compact, convex set with nonempty interior in $\mathbb{R}^n$. Then, given any $m$-dimensional subspace $F$ of $\mathbb{R}^n$, an application of Fubini's theorem yields
$$ \text{vol}_n(K) = \int_{P_{F^{\perp}}(K)} \text{vol}_m(K \cap (x + F)) dx, $$ where $P_{F^{\perp}}(K)$ denotes the orthogonal projection of $K$ onto the orthogonal complement $F^{\perp}$ of $F$, where $\text{vol}_m$ denotes the $m$-dimensional volume.
I really don't understand how the author got this identity by applying Fubini's theorem. Does anyone have any idea how this might work? Thanks!
The volume of $K$ can be written as the integral of the volumes/areas of $m$-dimensional cross-sections. Each cross-section $K \cap (x + F)$ can be indexed by an element $x \in P_{F^\perp}(K)$, as $x$ describes how far you need to slide the subspace $F$ to obtain the slice of $K$ that you want.
It may help to try examples with $n=2$ and $m=1$ (or $n=3$ and $m =2$) for intuition.