A Generalization of the Convex Kakeya Problem

TL;DR

This paper generalizes the convex Kakeya problem by characterizing optimal convex regions for containing translated segments, providing efficient algorithms for area minimization, and exploring perimeter minimization and containment of rotated figures.

Contribution

It proves the existence of a minimal-area convex triangle containing translated segments and offers an (n log n) algorithm to find it, extending Kakeya problem insights.

Findings

Optimal convex region is always a triangle.

Efficient (n log n) algorithm for minimal-area triangle.

For perimeter minimization, convex hull of midpoints is optimal.

Abstract

Given a set of line segments in the plane, not necessarily finite, what is a convex region of smallest area that contains a translate of each input segment? This question can be seen as a generalization of Kakeya's problem of finding a convex region of smallest area such that a needle can be rotated through 360 degrees within this region. We show that there is always an optimal region that is a triangle, and we give an optimal \Theta(n log n)-time algorithm to compute such a triangle for a given set of n segments. We also show that, if the goal is to minimize the perimeter of the region instead of its area, then placing the segments with their midpoint at the origin and taking their convex hull results in an optimal solution. Finally, we show that for any compact convex figure G, the smallest enclosing disk of G is a smallest-perimeter region containing a translate of every rotated copy of G.