This question is on the continuation of the post Approximation of convex body by polytopes
The central problem I am interested is an explicit construction of inscribed polytope with at most $n$ facets that can approximate a given convex body within Nikodym metric or Hausdorff metric.
In the recent Approximation of convex sets by polytopes, it seems that most general result are concerned with: (1) inscribed polytopal approximation with at most $n$ vertices; (2) circumscribed polytopal approximation with at most $n$ facets, but thes case for (3) inscribed polytopal approximation with at most $n$ facets; (4) circumscribed polytopal approximation with at most $n$ vertices are not addressed very well in the above survey.
It is claimed in Section 4.1 that all (1)-(4) can approximate any convex body in Hausdorff metric well, see (6) page 729: $$\inf_{P\in (i)}d_H(U,P)\leq \frac{c_U}{n^{2/(d-1)}}$$ for all $i=1,2,3,4$. However the literature there only refers to the proof for (1) and (2). My concerned would be mainly in (3), so my question is
(1) Is there an explicit and constructive proof for the case (3)?
(2) An even stronger assertion: For given convex body $U$, can we construct a sequence of polytope $P_1\subset P_2\cdots,$ so that $P_{n+1}\setminus P_{n}$ is union of polytopes(simplices) so that $d_H(U,P_n)\lesssim n^{-2/(d-1)}$?
