Algorithm 1024: Spherical Triangle Algorithm: A Fast Oracle for Convex Hull Membership Queries

The Convex Hull Membership (CHM) tests whether \( p \in conv(S) \), where p and the n points of S lie in \( \mathbb { R}^m \). CHM finds applications in Linear Programming, Computational Geometry, and Machine Learning. The Triangle Algorithm (TA), previously developed, in \( O(1/\varepsilon ^2) \) iterations computes \( p^{\prime } \in conv(S) \), either an \( \varepsilon \)- approximate solution , or a witness certifying \( p \not\in conv(S) \). We first prove the equivalence of exact and approximate versions of CHM and Spherical -CHM, where \( p=0 \) and \( \Vert v\Vert =1 \) for each v in S . If for some \( M \ge 1 \) every non-witness with \( \Vert p^{\prime }\Vert \gt \varepsilon \) admits \( v \in S \) satisfying \( \Vert p^{\prime } - v\Vert \ge \sqrt {1+\varepsilon /M} \), we prove the number of iterations improves to \( O(M/\varepsilon) \) and \( M \le 1/\varepsilon \) always holds. Equivalence of CHM and Spherical-CHM implies Minimum Enclosing Ball (MEB) algorithms can be modified to solve CHM. However, we prove \( (1+ \varepsilon) \)-approximation in MEB is \( \Omega (\sqrt {\varepsilon }) \)-approximation in Spherical-CHM. Thus, even \( O(1/\varepsilon) \) iteration MEB algorithms are not superior to Spherical-TA. Similar weakness is proved for MEB core sets. Spherical-TA also results a variant of the All Vertex Triangle Algorithm (AVTA) for computing all vertices of \( conv(S) \). Substantial computations on distinct problems demonstrate that TA and Spherical-TA generally achieve superior efficiency over algorithms such as Frank–Wolfe, MEB, and LP-Solver.