Imre Bárány, Lecture 4
نویسنده
چکیده
Let X ⊂ R d be a set of n points in general position (i.e. no d + 1 points of X lie on the same hyperplane.) For a positive 0 < ε < 1, we define A finite set S ⊂ R d is called a (weak) ε-net of X if S ∩ F = ∅ for any F ∈ F ε. We note that S is called a strong ε-net if, in addition, it also satisfies S ⊂ X. Since we are only going to discuss weak ε-nets, we will simply call these ε-nets. Our goal is to find an ε-net of X with small cardinality. Theorem 1. For any X ⊂ R d and any 0 < ε < 1, there exists an ε-net S of X with cardinality |S| ≤ c d 1 ε d+1 , where c d is a constant depending only on the dimension d. Note that this is a striking result: the cardinality of S does not depend on the set X! Proof. We may assume that ε > 2 d+1 n since, otherwise, 2 d+1 (d + 1) 1 ε d+1 ≥ 2 d+1 (d + 1) n d+1 2 d+1 (d + 1) d+1 ≥ n.
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تاریخ انتشار 2015