Finding a largest empty convex subset in space is W[1]-hard

TL;DR

This paper proves that finding the largest empty convex subset in space is computationally hard (W[1]-hard), indicating significant complexity in solving this geometric problem efficiently.

Contribution

The paper establishes the W[1]-hardness of the decision problem for finding the largest empty convex subset in three-dimensional space.

Findings

Proves the problem is W[1]-hard in computational complexity.

Highlights the difficulty of efficiently solving the problem in general cases.

Provides a complexity classification for a geometric problem.

Abstract

We consider the following problem: Given a point set in space find a largest subset that is in convex position and whose convex hull is empty. We show that the (decision version of the) problem is W[1]-hard.