Is it in NP to check if the convex hull contains the unit ball?
The problem is NP-hard; see my answer at mathoverflow. Thus there is no polynomial-size certificate that the unit ball is contained in the convex hull of given points unless $\text{NP} = \text{co-NP}$ (if $\text{NP} = \text{co-NP}$ then the polynomial hierarchy collapses).