Otfried Cheong, A Generalization of Kakeya’s Problem

One version of Kakeya’s problem asks for the smallest convex garden in which a ladder of length one can be rotated fully. The answer to this question was shown to be the equilateral triangle of height one by Pal in 1920.</p>

We consider the following generalization: Given a (not necessarily finite) family F of line segments in the plane, what is the smallest convex figure K such that every segment in F can be translated to lie in K?

We show that as in the classic case, the answer can always be chosen to be a triangle. We also give an O(n log n) time algorithm to compute such an optimal triangle if F consists of n line segments.

Joint work with Sang Won Bae, Hee-Kap Ahn, Joachim Gudmundsson, Takeshi Tokuyama, and Antoine Vigneron.