Plank problems: discrete geometry and convexity

A plank in a Euclidean space is the region between two parallel hyperplanes, and its width is the distance between these hyperplanes. What is the smallest total width of planks needed to cover a given convex region in the plane? What happens in higher dimensions? In the 1950s, Thøger Bang answered this innocent-looking question, posed by Alfred Tarski, by proving that the total width must be at least the width of the thinnest plank containing the region. His result opened the door to many deceptively simple-looking problems.

In this talk, I will survey progress on plank problems and discuss their connections with theoretical computer science, number theory, and analysis. In particular, I will present joint work with Zilin Jiang confirming Fejes Tóth’s long-standing zone conjecture, as well as results with Alexey Glazyrin and Roman Karasev on a polynomial plank problem --- a far-reaching generalization of Bang’s theorem.

Finally, I will discuss very recent joint work with Egor Bakaev on optimal partial plank coverings. If the total available width is fixed, how should the planks be arranged to cover the largest possible portion of a convex body? We show that, for Euclidean balls in every dimension and for all planar convex bodies, an optimal arrangement consists of a single plank.
 

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