# Convex Hull of $\left(t, t^2, \cdots, t^N\right)$

## TL;DR

This paper characterizes the convex hull of the curve (t, t^2, ..., t^N), showing it can be described by convex combinations of (N+1)/2 points and establishing a homeomorphism between these combinations and the hull.

## Contribution

It provides a detailed description of the convex hull of the parametric curve and proves the convex combination representation and a homeomorphism for the convex hull.

## Key findings

- Every point in the convex hull is a convex combination of (N+1)/2 points on the curve.
- The convex combination evaluation is a homeomorphism onto the convex hull.
- The representation is invariant under certain operations on the convex combinations.

## Abstract

This paper analyzes the convex hull of the parametric curve $\left(t, t^2, \cdots, t^N\right)$, where $t$ is in a closed interval. It finds that every point in the convex hull is representable as a convex combination of $\frac{N+1}{2}$ points on the curve. It also finds that the evaluation of the convex combination is a homeomorphism from the convex combinations of $\frac{N+1}{2}$ 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.