Convex Covers and Translation Covers

ABSTRACT:
In 1914, Lebesgue asked for a convex set of smallest possible area that can contain a congruent copy of every set of diameter one. The same question can be asked for other families T of planar shapes: What is the convex set of smallest possible area that contains a congruent copy of every element of T? Such a set is then called a convex cover for T, and we will see what smallest-area convex covers for some families of triangles look like. A translation cover for a family T of planar shapes is defined similarly: Z is a translation cover for T if every element of T can be translated into Z. Kakeya's celebrated needle problem, first posed in 1917, turns out to be a question about a smallest-area translation cover. We will see that the generalization of Kakeya's problem to other shapes is also a translation cover problem.

BIOGRAPHY:
Otfried Cheong received his Ph.D. at FU Berlin in 1992. After holding positions at Utrecht University, Postech, Hong Kong University of Science & Technology, and TU Eindhoven, he has been at KAIST since 2005. He is on the editorial board of 'Discrete & Computational Geometry' and 'Computational Geometry: Theory & Applications', and was elected an ACM Distinguished Scientist in 2016. He is currently on leave from KAIST to work with Scalgo on water flow simulations.