What is the convex hull of the rational points of the unit sphere? Here, rational points are points whose coordinates are rational numbers.
Also, what is the convex hull of the irrational points of the unit sphere? Here, irrational points are points that are not rational points.
I am looking for any properties that could help visualize the thing in question. For instance, is it smooth or not? how smooth? how small is it compared to the sphere, say, area- or volume-wise?
Motivation: Cauchy's theorem says that convex polyhedra with the same net are congruent. There is a generalization, due to Pogorelov, that says that convex surfaces with the same metric are congruent. I am trying to understand how impressive this generalization is by trying to visualize how nasty (compact) convex surfaces can be compared to a convex polyhedron. The convex hulls above are the nastiest I could think of. Hence the question.
In both cases they are the entire interior of the unit sphere, plus exactly the [rational/irrational] points on the boundary.
To see that the whole interior is obtained, observe that any point strictly within the interior of the sphere is contained within the interior of some finite spherical polyhedron, and then we can choose nearby rational or irrational points as necessary without affecting containment.
To see that no other points on the boundary besides what we started with get included, consider a point $P$ not in our generating set, and the closed half-space $H$ tangent to the sphere containing only $P$. Since the complement of $H$ is a convex set and our generating set is a subset of it, the convex hull of that set will also be a subset of it (hence won't contain $P$).