弱ε-网的下界与阶梯凸性
Lower bounds for weak epsilon-nets and stair-convexity
摘要：
对于R^d中的有限集X，如果N与X的至少包含X * |X|点的每个凸集相交，则集合N被称为“弱ε-网”（相对于凸集）。对于每个固定的d>=2和r>=1，我们在R^d中构造集合X，使得每个弱（1/r）-网至少有Omega（r log^{d-1} r）个点;这是固定维弱ε-网的第一个超线性下界。该结构是一个“拉伸网格”，即，d个合适的快速增长的有限序列的笛卡儿积，并且该网格中的凸性可以使用“阶梯凸性”来分析，“阶梯凸性”是凸性的通常概念的新变体。我们还考虑了在R^d，d>=3中拉伸网格的对角线上的弱ε-网，这是一个“本质上1维”的点集。在这种情况下，我们展示了轻微的超线性下界（涉及逆阿克曼函数），表明Alon，Kaplan，Nivasch，Sharir和Smorodinsky（2008）的上界在最坏情况下离真理不远。使用拉伸网格，我们还通过对数因子改进了平面中所谓的“第二选择引理”的已知上界：我们得到一个t个三角形的集合T，其顶点在平面中的一个n点集合中，使得没有点包含在T的O（t^2 /（n^3 log（n^3/t）个三角形中。
英文摘要：
A set N is called a "weak epsilon-net" (with respect to convex sets) for a finite set X in R^d if N intersects every convex set that contains at least epsilon*|X| points of X. For every fixed d>=2 and every r>=1 we construct sets X in R^d for which every weak (1/r)-net has at least Omega(r log^{d-1} r) points; this is the first superlinear lower bound for weak epsilon-nets in a fixed dimension. The construction is a "stretched grid", i.e., the Cartesian product of d suitable fast-growing finite sequences, and convexity in this grid can be analyzed using "stair-convexity", a new variant of the usual notion of convexity. We also consider weak epsilon-nets for the diagonal of our stretched grid in R^d, d>=3, which is an "intrinsically 1-dimensional" point set. In this case we exhibit slightly superlinear lower bounds (involving the inverse Ackermann function), showing that upper bounds by Alon, Kaplan, Nivasch, Sharir, and Smorodinsky (2008) are not far from the truth in the worst case. Using the stretched grid we also improve the known upper bound for the so-called "second selection lemma" in the plane by a logarithmic factor: We obtain a set T of t triangles with vertices in an n-point set in the plane such that no point is contained in more than O(t^2 / (n^3 log (n^3/t))) triangles of T.
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