Let $S \subset \mathbb{R}^n$ be some "nice" bounded set whose diameter isn't "too small". We can define the measure $$ \nu (S) = \sum_{m \in \mathbb{Z}} \mathrm{Leb}\left( \lbrace \mathbf{x} \in \mathbb{R}^n : x_1 = m\rbrace \cap S \right) $$ where $\mathrm{Leb}$ is $(n-1)$-dimensional Lebesgue measure, and $x_1$ is the first coordinate of $\mathbf{x}$. In two dimensions, $\nu$ measures the lengths of various slices of $S$ and adds them up. In three dimensions, it's about the areas of various cross-sections, and so on.
Now, $\nu(S)$ is a Riemann sum with step length $1$ for the integral $\int_{\mathbb{R}^n} I_S \, \mathrm{d}\mathbf{x} = \mathrm{Vol} (S)$, and hence it's reasonable to expect $\nu (S) \approx \mathrm{Vol}(S)$ when $\mathrm{diam} \, S$ isn't too small. The volume of $S$ is invariant when $S$ is rotated, i.e.
$$ \mathrm{Vol}(KS) = \mathrm{Vol}(S), \quad K \in SO(n), $$
but in general $\nu$ will not be invariant under this action. Can we say anything about how much $\nu (KS)$ varies as $K$ runs over $SO(n)$ (in terms of the diameter or other geometric attributes of $S$)?
I suspect this problem is quite hard in general. If you are aware of any references that discuss this question, or something along these lines, I would be very interested in finding out about them!