Title: A Generalization of the Convex Kakeya Problem.
Speaker: Hee-Kap Ahn, POSTECH, Pohang, South Korea
Date: September 23, 2014
Abstract:
A classical translation cover problem is the Kakeya needle problem. It
asks for a minimum area convex region in which a needle of length 1
can be rotated through 360 degrees continuously and return to its
initial position. This problem was posed by Soichi Kakeya in 1917, and
P\`{a}l showed that the solution of the problem is the equilateral
triangle of height one.
We consider a generalization of the Kakeya needle problem that, given
a set of line segments in the plane, not necessarily finite, asks for
a minimum area convex region containing a translate of each input
segment. We show that there is always an optimal region that is a
triangle, and we give an optimal algorithm to compute such a triangle
for a given set of segments. We also show that, if the goal is to find
a convex region of smallest perimeter, an optimal solution can be
obtained by placing the segments with their midpoint at the origin and
taking their convex hull results.
Joint work with Sang Won Bae, Otfried Cheong, Joachim Gudmundsson,
Takeshi Tokuyama, Antoine Vigneron