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By K. Ribnikov

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Second, let us recall that if X is a finite (point) set lying in the interior of a unit ball in Ed , then the intersection of the (closed) unit balls of Ed centered at the points of X is called a ball-polyhedron and it is denoted by B[X]. ) Of course, it also makes sense to introduce B[X] for sets X that are not finite but in those cases we get sets that are typically not ball-polyhedra. Now, we are ready to state our theorem. 4. Let B[X] ⊂ Ed be a ball-polyhedron of minimal width x with 1 ≤ x < 2.

5. Let C be a convex body in Ed and let m be a positive integer. Then let TCm,d denote the family of all sets in Ed that can be obtained as the intersection of at most m translates of C in Ed . 6. Let C be a convex body of minimal width w > 0 in Ed and let 0 < x ≤ w be given. Then for any non-negative integer n let n vd (C, x, n) := min{vold (Q) | Q ∈ TC2,d and w(Q) ≥ x}. Now, we are ready to state the theorem which although was not published by Bang in [8], it follows from his proof of Tarski’s plank conjecture.

It is quite remarkable that Tarski’s question and its variants continue to generate interest in the geometric and analytic aspects of coverings by planks in the present time as well. The paper is of a survey type with some new results and with a list of open research problems on the discrete geometric side of the plank problem. 1. Introduction Tarski’s plank problem has generated a great interest in understanding the geometry of coverings by planks. There have been a good number of results published in connection with the plank problem of Tarski that are surveyed in this paper.

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