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The Radon number of the three-dimensional integer lattice. (English) Zbl 1043.52010

S. Onn [SIAM J. Discrete Math. 4, No. 3, 436–447 (1991; Zbl 0735.52007)] proved that the Radon number \(r(d)\) of the \(d\)-dimensional integer lattice fulfills the inequalities \(5\cdot 2 ^{d-2} +1 \leq r(d) \leq d(2^ d -1) +3\). So in particular, \(11 \leq r(3) \leq 24\). The authors of the present note improve the right inequality up to \(r(3) \leq 17\).

MSC:

52B20 Lattice polytopes in convex geometry (including relations with commutative algebra and algebraic geometry)
68Q25 Analysis of algorithms and problem complexity
52A35 Helly-type theorems and geometric transversal theory

Citations:

Zbl 0735.52007
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