Convex Hull of (t, t², ⋯, t^N)

2017/06/07 by Kostyantyn Mazur, Mazur, Kostyantyn
Engineering · Mathematics · #Advanced Numerical Analysis Techniques #FOS: Mathematics #General Topology (math.GN) #Mathematics and Applications #Point processes and geometric inequalities

paper · pdf · doi:10.48550/arxiv.1706.02060

openalex publication_date 2017/06/07 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

This paper analyzes the convex hull of the parametric curve (t, t², ⋯, t^N), where t is in a closed interval. It finds that every point in the convex hull is representable as a convex combination of (N+1)/(2) points on the curve. It also finds that the evaluation of the convex combination is a homeomorphism from the convex combinations of (N+1)/(2) points on the curve to the convex hull of the curve, as long as the points are listed in increasing order, and as long as two representations that are reachable from each other by removing terms with coefficient zero, combining terms with the same point, the inverses of these operations, or a sequence of these operations in any order, are considered to be equivalent.

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