Lebesgue's differentiation theorem states that if $x$ is a point in $\mathbb{R}^n$ and $f:\mathbb{R}^n\rightarrow\mathbb{R}$ is a Lebesgue integrable function, then the limit of $\frac{\int_B f d\lambda}{\lambda(B)}$ over all balls $B$ centered at $x$ as the diameter of $B$ goes to $0$ is equal almost everywhere to $f(x)$. But if you replace balls with other kinds of set with diameter going to $0$, this need not be true. For instance it need not be true if you replace balls with rectangles.

It does hold for squares, though, and more generally for $n$-dimensional cubes. So my question is, is it known if it holds for all regular polygons? And more generally, is it known if it hold for all regular polyhedra and regular polytopes?

I ask "is it known" because a lot of problems in this topic tend to be open.