MINVO Basis: Finding Simplexes with Minimum Volume Enclosing Polynomial Curves

This paper studies the polynomial basis that generates smallest n-simplex enclosing a given nth-degree curve in Rn. Although Bernstein and B-Spline bases provide feasible solutions to this problem, simplexes obtained by these are not possible, which leads overly conservative results many CAD (computer-aided design) applications. We first prove solves problem (MINVO basis) also for with largest convex hull enclosed n-simplex. Then, we present formulation is independent of or given. By using Sum-Of-Squares (SOS) programming, branch bound, moment relaxations, obtain high-quality any n∈N, (numerical) global optimality n=1,2,3 local n=4. The n=3 show that, 3rd-degree R3, MINVO able an simplex whose volume 2.36 254.9 times smaller than ones bases, respectively. When n=7, ratios increase 902.7 2.997⋅1021,