Title: A generalization of Kakeya’s problem

Speaker: Otfried Cheong, KAIST

Time/Date: Monday, May 23, 11-12
Location: Room 3501

Abstract:

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.

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.