The largest convex set in the interior of a non-convex set We can define the convex hull, which is the smallest convex set containing a set $A$. I believe this can be viewed, essentially, as approximating any set from the outside via convex sets. Is there an analogous notion for approximating the inside of a non-convex set? In the sense that we may find the 'largest' convex set contained within a non-convex set?
 A: There is a fundamental issue here.  The convex hull of a set $A$ is the intersection of all convex sets containing $A$.  This definition "works", or "makes sense", because the intersection of convex sets is itself convex.
So to ask a corresponding question "from the inside", you would be looking at the union of all convex sets contained in $A$.  Problem is, the union of convex sets is generally speaking not convex.
You could ask something like "what is the largest (in volume or area) convex set contained in $A$?" and it might be possible to answer this.  However it would be fundamentally different from the usual "convex hull" question, and you could not reasonably expect any meaningful connections between the two problems.
In particular, if you take $A$ to be, for example, an annulus, then it is intuitively clear that no convex set within $A$ is going to be a "good approximation" to $A$.
A: The set of all rational numbers does not contain any convex set except single points so there is no convex susbet that is 'close' to this set. 
