By Imre Bárány, Károly Jr. Böröczky, Gábor Fejes Tóth, Janos Pach

The current quantity is a suite of a dozen survey articles, devoted to the reminiscence of the well-known Hungarian geometer, László Fejes Tóth, at the 99th anniversary of his beginning. each one article studies fresh development in an incredible box in intuitive, discrete, and convex geometry. The mathematical paintings and views of all editors and such a lot participants of this quantity have been deeply stimulated through László Fejes Tóth.

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10] A. Bezdek and K. Bezdek, Conway’s fried potato problem revisited, Arch. Math. 66/6 (1996), 522–528. [11] A. Bezdek, Covering an annulus by strips, Discrete Comput. Geom. 30 (2003), 177– 180. [12] A. Bezdek, On a generalization of Tarski’s plank problem, Discrete Comput. Geom. 38 (2007), 189–200. [13] K. Bezdek and T. Hausel, On the number of lattice hyperplanes which are needed to cover the lattice points of a convex body, Colloq. Math. Soc. J´´anos Bolyai 63 (1994), 27–31. [14] K. Bezdek, R.

1. Definitions and Preliminaries A d-dimensional convex body is a compact convex subset of Rn , contained in a d-dimensional flat and with non-void interior relative to the flat. A 2dimensional convex body is called a convex disk. The (d-dimensional) volume of a d-dimensional convex body K will be denoted by Vol(K), but ∗ Both authors gratefully acknowledge research support: A. Bezdek was supported by the Hungarian Research Foundation OTKA, grant #068398; W. Kuperberg was supported in part by the DiscConvGeo (Discrete and Convex Geometry) project, in the framework of the European Community’s “Structuring the European Research Area” programme.

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. 7. Let C be a convex body of minimal width w > 0 in Ed . Moreover, let P1 , P2 , . . , Pn be planks of width w1 , w2 , . . , wn in Ed with w0 = w1 + w2 + · · · + wn < w. Then vold (C \ (P1 ∪ P2 ∪ · · · ∪ Pn )) ≥ vd (C, w − w0 , n), that is vold ((P1 ∪ P2 ∪ · · · ∪ Pn ) ∩ C) ≤ vold (C) − vd (C, w − w0 , n). Clearly, the first inequality above implies (via an indirect argument) that if the planks P1 , P2 , .

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