Algorithms for a Minimum Volume Enclosing Simplex in Three Dimensions
Yunhong Zhou, Subhash Suri · SIAM Journal on Computing · 2002
We develop a combinatorial algorithm for determining a minimum volume simplex enclosing a set of points in ${\cal R}^3$. If the convex hull of the points has n vertices, then our algorithm takes $\Theta(n^4)$ time. Combining our exact but slow algorithm with a simple but crude approximation technique, we also develop an $\varepsilon$-approximation algorithm. The algorithm computes in $O(n + 1/\varepsilon^6)$ time a simplex whose volume is within $(1 + \varepsilon)$ factor of the optimal for any $\varepsilon > 0$.