Erschienen in:

2003 | OriginalPaper | Buchkapitel

Integer Points in Rotating Convex Bodies

verfasst von : Imre Bárány, Jiří Matoušek

Erschienen in: Discrete and Computational Geometry

Verlag: Springer Berlin Heidelberg

Enthalten in: Professional Book Archive

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Let K be a planar convex body symmetric about the origin. We define P(K) as the probability of τ K∩Z2 ≠ {0}, where τ ∈ SO(2) is a random rotation around the origin and SOZ2 denotes the integer lattice, and we let P(v) = inf{P(K) : vol(K) = υ}. By Minkowski’s theorem, P(υ) = 1 for υ > 4, and P(υ) = 0 for υ < π. We describe the behavior of P(υ) in the intervals [π, π + ε₀] and [4 - ε₀, 4] for a small positive constant ε₀.

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