Tarski's plank problem

In mathematics, Tarski's plank problem is a question about coverings of convex regions in n-dimensional Euclidean space by "planks": regions between two hyperplanes.

The question was answered affirmatively by Thøger Bang (1950, 1951).

The (closed) set of points P between two distinct, parallel hyperplanes in Rn is called a plank, and the distance between the two hyperplanes is called the width of the plank, w(P).

Tarski conjectured that if a convex body C of minimal width w(C) was covered by a collection of planks, then the sum of the widths of those planks must be at least w(C).

Bang's theorem shows that, for example, a circular table of diameter d feet can't be covered by fewer than d planks of wood of width one foot each.