  Let $K$ be a convex body in $\Bbb R^{d}$ and $K_{t}$ its floating bodies.
There is a polytope with at most $n$ vertices that satisfies $$ K_{t} \subset
P_{n} \subset K $$ where $$ n \leq e^{16d} \frac{vol_{d}(K \setminus K_{t})}{t\
vol_{d}(B_{2}^{d})} $$ Let $K^{t}$ be the illumination bodies of $K$ and
$Q_{n}$ a polytope that contains $K$ and has at most $n$ $d-1$-dimensional
faces. Then $$ vol_{d}(K^{t} \setminus K) \leq cd^{4} vol_{d}(Q_{n} \setminus
K) $$ where $$ n \leq \frac{c}{dt} \ vol_{d}(K^{t} \setminus K) $$
