This is a question-by-proxy for a colleague from computer science. I'm sure many people here are already aware that *convex decomposition* forms an important sub-field of both computational geometry and vision (in the CS sense). The prototypical problem in this field is as follows:

Consider a nice compact, connected set $B \subset \mathbb{R}^n$ with non-zero volume (this set is typically a polyhedron). Find a minimal collection of closed convex subsets $C_j \subset B$ such that the union of $C_j$'s equals $B$, and all pairwise intersections $C_i \cap C_j$ for $i \neq j$ have zero volume.

A lot of work has gone into this, presumably because convex objects are considered simpler to analyze than arbitrary compact blobs. I'm no expert in the field, but I believe it is quite standard: algorithms have been implemented in CGAL, based on Chazelle's pioneering work.

A lot of the theory breaks down almost immediately if one starts to step away from polyhedra. My question here can be best explained with this picture:

Note that there is no convex decomposition for the square minus disc on the left. So,

Which compact sets with non-zero volume are definable as

finite unions and intersections of convex sets and their complements?

And of course, the natural follow-up: if we allow *countable* rather than finite unions and intersections and restrict our domain to some compact, convex subset of $\mathbb{R}^n$, then the definable sets naturally generate a sigma algebra.

Is this convex sigma algebra the same as the traditional one with open sets as a basis?

On one hand, every open ball is a countable union of closed convex subsets. But what about the other direction?