Abstract: The set of points lying between two parallel lines in the plane is called a plank. Given a convex region C in the plane, what is the smallest combined width of a finite collection of planks that cover C? What happens in higher dimensions? In the 50s, Thoger Bang answered this innocent question of Alfred Tarski and opened a box with many deceptively simple-looking problems.

In my talk, I will overview progress in the area and its connection with other fields: theoretical computer science, number theory, and analysis. In particular, I will discuss a joint work with Zilin Jiang confirming Fejes Toth’s long-standing zone conjecture and recent results with Alexey Glazyrin and Roman Karasev on a polynomial plank problem, a far-reaching generalization of Bang’s theorem.