A Polynomial-Time Algorithm for Computing the Exact Convex Hull in High-Dimensional Spaces

This study presents a novel algorithm for identifying the set of extreme points that constitute the exact convex hull of a point set in high-dimensional Euclidean space. The proposed method iteratively solves a sequence of dynamically updated quadratic programming (QP) problems for each point and exploits their solutions to provide theoretical guarantees for exact convex hull identification. For a dataset of $n$ points in an $m$-dimensional space, the algorithm achieves a dimension-independent worst-case time complexity of $O(n^{p+2} \log(1/\epsilon))$, where $p$ depends on the choice of QP solver (e.g., $p = 4$ corresponds to the worst-case bound when using an interior-point method), and $\epsilon$ denotes the target numerical precision (i.e., the optimality tolerance of the QP solver). The proposed method is applicable to spaces of arbitrary dimensionality and exhibits particular efficiency in high-dimensional settings, owing to its polynomial-time complexity, whereas existing exponential-time algorithms become computationally impractical.