An Algorithm for Computing the Exact Convex Hull in High-Dimensional Spaces

This study introduces a novel algorithm for computing the convex hull of a point set in high-dimensional Euclidean space. The method iteratively solves a sequence of dynamically updated quadratic programming (QP) problems for each point and leverages the solutions to provide theoretical guarantees for exact convex hull identification. Given $n$ points in an $m$-dimensional space, the algorithm achieves a best-case time complexity of $O(nm^p)$, where $p$ depends on the specific QP solver employed (e.g., an interior-point method). In the worst case, the complexity increases to $O(n h^{p+1})$, where $h$ denotes the minimal number of vertices defining the convex hull. The approach is particularly well-suited for large-scale, high-dimensional datasets, where exponential-time algorithms become computationally infeasible.