# Computing a Minimum-Width Cubic and Hypercubic Shell

## TL;DR

This paper introduces the first efficient algorithm for computing the minimum-width axis-aligned cubic shell enclosing a set of points, with extensions to higher dimensions, achieving near-optimal running times.

## Contribution

It presents the first known algorithm for the minimum-width cubic shell problem, with a running time of O(n log^2 n), and extends the approach to higher dimensions.

## Key findings

- First nontrivial algorithm for cubic shell problem
- Algorithm runs in O(n log^2 n) time in 3D
- Extension to higher dimensions with O(n^{⌊d/2⌋} log^{d-1} n) complexity

## Abstract

In this paper, we study the problem of computing a minimum-width axis-aligned cubic shell that encloses a given set of $n$ points in a three-dimensional space. A cubic shell is a closed volume between two concentric and face-parallel cubes. Prior to this work, there was no known algorithm for this problem in the literature. We present the first nontrivial algorithm whose running time is $O(n \log^2 n)$. Our approach easily extends to higher dimension, resulting in an $O(n^{\lfloor d/2 \rfloor} \log^{d-1} n)$-time algorithm for the hypercubic shell problem in $d\geq 3$ dimension.