MinimumConvexHull

MinimumConvexHull is the smallest convex set that contains a given finite set of points in a Euclidean space. In two dimensions, it corresponds to the convex hull: a convex polygon whose vertices are a subset of the input points and which encloses all points. In higher dimensions, it is a convex polytope whose facets bound the minimal convex region containing the data. The convex hull is unique and serves as the tight boundary around the point set.

Computationally, several well-established algorithms construct MinimumConvexHull. In the plane, Graham scan and Andrew’s monotone chain run in O(n log n) time, while the gift-w wrapping (Jarvis march) algorithm has a time complexity of O(nh), where h is the number of hull vertices. In higher dimensions, algorithms such as Quickhull, Beneath-Beyond, and Chan’s algorithm are used, with efficiencies that depend on the dimension and input size. The hull is typically output as an ordered sequence of its vertices or as a set of supporting facets.

Applications of MinimumConvexHull span numerous fields. It is used in pattern recognition, computer graphics, collision detection, geographic information systems, clustering, and shape analysis as a compact geometric summary of a point cloud or dataset. It also serves as a building block for more complex geometric constructs, such as bounding volumes and proximity queries.

Practical considerations include handling degenerate input (e.g., many collinear points or duplicate points) and numerical robustness. Implementations rely on careful geometric predicates and, in some cases, exact arithmetic or epsilon-based tolerance to mitigate floating-point errors. Many numerical libraries expose convex-hull routines under a name akin to MinimumConvexHull.