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By V. I Arnolʹd

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2 If the convex body C is covered by the planks P1 , P2 , . . , Pn in Ed , d ≥ 2, then n wC (Pi ) ≥ 1. 2, when the convex body to be covered is centrally symmetric, has been proved by Ball in [12]. Thus, the following is Ball’s plank theorem. 3 If the centrally symmetric convex body C is covered by the planks P1 , P2 , . . , Pn in Ed , d ≥ 2, then n wC (Pi ) ≥ 1. 1 to derive a new argument for an improvement of the Davenport–Rogers lower bound on the density of economical sphere lattice packings.

32 3 Coverings by Homothetic Bodies - Illumination and Related Topics Kiss and de Wet [180] conjecture the following. 3 The illumination parameter of any o-symmetric convex body in E3 is at most 12. Motivated by the notion of the illumination parameter Swanepoel [235] introduced the covering parameter cov(Ko ) of Ko in the following way. (1 − λi )−1 | Ko ⊂ ∪i (λi Ko + ti ), 0 < λi < 1, ti ∈ Ed }. cov(Ko ) = inf{ i In this way homothets almost as large as Ko are penalised. Swanepoel [235] proved the following fundamental inequalities.

Ln such that Li ⊂ Ed \ K for all 1 ≤ i ≤ n and L1 , L2 , . . , Ln illuminate K. Il (K) is called the l-dimensional illumination number of K and the sequence I0 (K), I1 (K), . . , Id−2 (K), Id−1 (K) is called the successive illumination numbers of K. Obviously, I0 (K) ≥ I1 (K) ≥ · · · ≥ Id−2 (K) ≥ Id−1 (K) = 2. Let Sd−1 be the unit sphere centered at the origin of Ed . Let HS l ⊂ Sd−1 be an l-dimensional open great-hemisphere of Sd−1 , where 0 ≤ l ≤ d − 1. Then HS l illuminates the boundary point q of K if there exists a unit vector v ∈ HS l that illuminates q, in other words, for which it 30 3 Coverings by Homothetic Bodies - Illumination and Related Topics is true that the halfline emanating from q and having direction vector v intersects the interior of K.

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Contact geometry and wave propagation: Lectures given at the University of Oxford under the sponsorship of the International Mathematical Union by V. I Arnolʹd


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