Abstract: This paper studies the problem of finding the smallest n-simplex enclosing a given nth-degree polynomial curve. Although the Bernstein and B-Spline polynomial bases provide feasible solutions to this problem, the simplexes obtained by these bases are not the smallest possible, which leads to undesirably conservative results in many applications. We first prove that the polynomial basis that solves this problem (MINVO basis) also solves for the nth-degree polynomial curve with largest convex hull enclosed in a given n-simplex. Then, we present a formulation that is \emph{independent} of the n-simplex or nth-degree polynomial curve given. By using Sum-Of-Squares (SOS) programming, branch and bound, and moment relaxations, we obtain high-quality feasible solutions for any n∈ℕ and prove numerical global optimality for n=1,2,3. The results obtained for n=3 show that, for any given 3rd-degree polynomial curve, the MINVO basis is able to obtain an enclosing simplex whose volume is 2.36 and 254.9 times smaller than the ones obtained by the Bernstein and B-Spline bases, respectively. When n=7, these ratios increase to 902.7 and 2.997⋅1021, respectively.
Authors: Jesus Tordesillas, Jonathan P. How (MIT)