Algorithm that generates a n-simplex that cover n-polytope?

Question

Given an $n$-cube with unit volume, is there any algorithm that generates a $n$-simplex that covers the $n$-polytope?

Answer 1

There is a nontrivial algorithm in $\mathbb{R}^3$ for the minimum volume tetrahedron circumscribing an arbitrary polyhedron (i.e., not specifically a cube):

Zhou, Yunhong, and Subhash Suri. "Algorithms for a minimum volume enclosing simplex in three dimensions." SIAM Journal on Computing 31.5 (2002): 1339-1357. (ACM link.)
               

This work relies on a beautiful theorem of Victor Klee, valid in all dimensions, that implies touching at facet centroids (as one can see in the above figure):

V. Klee, Facet-centroids and volume minimization, Studia Sci.Math.Hungar., 21 (1986), pp.143–147.