# Classical topics in discrete geometry by K?roly Bezdek By K?roly Bezdek

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As is well known 1 ≤ him(K) ≤ d. Using this notion Boltyanski  gave a proof of the following theorem. 5 Let K be a convex body with him(K) = 2 in Ed , d ≥ 3. Then I(K) ≤ 2d − 2d−2 . In fact, in  Boltyanski conjectures the following more general inequality. 6 Let K be a convex body with him(K) = h > 2 in Ed , d ≥ 3. Then I(K) ≤ 2d − 2d−h . The author and Bisztriczky gave a proof of the Illumination Conjecture for the class of dual cyclic polytopes in . Their upper bound for the illumination numbers of dual cyclic polytopes has been improved by Talata in .

Wn in Ed such that their union covers the largest volume subset of C, that is, for which vold ((P1 ∪ P2 ∪ · · · ∪ Pn ) ∩ C) is as large as possible. As the following special case is the most striking form of the above problem, we are putting it forward as the main question of this section. 1 Let Bd denote the unit ball centered at the origin o in Ed . Moreover, let w1 , w2 , . . , wn be positive real numbers satisfying the inequality w1 + w2 + · · · + wn < 2. Then prove or disprove that the union of the planks P1 , P2 , .

3 (i) If K is a centrally symmetric convex body in E3 , then I(K) ≤ 8. (ii) If K is a convex body symmetric about a plane in E3 , then I(K) ≤ 8. Lassak  and later also Weissbach  and the author, L´angi, Nasz´odi, and Papez  gave a proof of the following. 4 The illumination number of any convex body of constant width in E3 is at most 6. It is tempting to conjecture the following even stronger result. If true, then it would give a new proof and insight of the well-known theorem, conjectured by Borsuk long ago (see for example ), that any set of diameter 1 in E3 can be partitioned into (at most) four subsets of diameter smaller than 1.