# A generalization of the convex kakeya problem

H.K. Ahn, S.W. Bae, O. Cheong, J. Gudmundsson, T. Tokuyama, A. Vigneron

Algorithmica, 70(2), 152-170, (2014)

## Keywords

Convex kakeya problem

## Abstract

We consider the following geometric alignment problem: Given a set of
line segments in the plane, find a convex region of smallest area that
contains a translate of each input segment. This can be seen as a
generalization of Kakeya’s problem of finding a convex region of
smallest area such that a needle can be turned through 360 degrees
within this region. Our main result is an optimal Θ(*n* log*n*)-time algorithm for our geometric alignment problem, when the input is a set of *n*
line segments. We also show that, if the goal is to minimize the
perimeter of the region instead of its area, then the optimum placement
is when the midpoints of the segments coincide. 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 any rotated copy of *G*.

## Code

DOI: 10.1007/978-3-642-29344-3_1