Plank theorems and their applications: a survey

Abstract

Plank problems concern the covering of convex bodies by planks in Euclidean space and are related to famous open problems in convex geometry. In this survey, we introduce plank problems and present surprising applications of plank theorems in various areas of mathematics. Tarski's plank problem The genesis of plank problemsIn the 1930s, the mathematician and logician Alfred Tarski, who is known for his work on model theory and the Banach-Tarski paradox, among other things, proposed a problem that would change the face of discrete geometry. It is what we now call a plank problem. Conjecture 2.1 (Tarski [53], 1932). If C ⊂ R d is covered by a sequence of planks P 1 , P 2 . . . , P n , then the sum of the widths of the planks is at least w(C).Without loss of generality, we may consider a body of minimal width 1. Then, similarly to our answer to Question 1.1, it is obvious that we can cover a convex body of minimal width 1 with planks that have total width 1, by placing them perpendicularly to the hyperplanes that support the convex body. But can we do better? Tarski's conjecture says no. Partial solution to Tarski's problemTarski proved his conjecture for figures in which we can inscribe a disk centered at the origin. Note that the case of R 2 is not entirely covered by this partial result, since some convex figures, for example an equilateral triangle, have a width bigger than the diameter of their inscribed circle; yet, Tarski's arguments generalize to solids in which we can inscribe a ball centered at the origin. However, they do not work in higher dimensions because the proof relies on geometric characteristics of two (or three)-dimensional spaces. His proof is still interesting in its own right, so we include it by adapting the argument that was presented in [35]. Note that Tarski was inspired by a solution of Moese on a related problem first stated by Tarksi himself (read [42, Chapter 7] for a discussion on the history of this problem and a translation of the related papers of Tarski and Moese).The demonstration relies essentially on a result of Archimedes in On the Sphere and Cylinder, often called "Archimedes' Hat-Box Theorem", which can be formulated as follows.Proposition 2.2. The lateral area of a spherical segment formed by the intersection of a sphere of radius r and two planes separated by distance d is 2πrd.