Cutting a Convex Polyhedron Out of a Sphere

TL;DR

This paper presents an algorithm that efficiently cuts a convex polyhedron from a sphere with near-optimal cutting cost, using a method that guarantees a logarithmic approximation factor.

Contribution

It introduces an $O(n^3)$-time algorithm for cutting out a convex polyhedron from a sphere with a provably near-optimal cost using guillotine cuts.

Findings

Algorithm runs in $O(n^3)$ time.

Cutting cost is within $O((\log n)^2)$ of optimal.

Provides a practical approach for geometric cutting problems.

Abstract

Given a convex polyhedron $P$ of $n$ vertices inside a sphere $Q$, we give an $O(n^3)$-time algorithm that cuts $P$ out of $Q$ by using guillotine cuts and has cutting cost $O((\log n)^2)$ times the optimal.