# Finding the vertices of the convex hull, even unordered, takes Omega(n log n) time -- a proof by reduction from epsilon-closeness

## TL;DR

This paper proves that identifying convex hull vertices in an unordered set of points requires at least Omega(n log n) time, using a reduction from epsilon-closeness, thus establishing a fundamental complexity bound.

## Contribution

It offers a simple reduction-based proof establishing the Omega(n log n) lower bound for convex hull vertex identification in the algebraic decision tree model.

## Key findings

- Convex hull vertex identification has an Omega(n log n) lower bound.
- The proof uses a reduction from epsilon-closeness problem.
- The result applies to unordered point sets in the plane.

## Abstract

We consider the problem of computing, given a set S of n points in the plane, which points of S are vertices of the convex hull of S. For certain variations of this problem, different proofs exist that the complexity of this problem in the algebraic decision tree model is Omega(n log n). This paper provides a relatively simple proof by reduction from epsilon-closeness.