dense point set
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2019 ◽  
Vol 64 (3) ◽  
pp. 965-984
Author(s):  
István Kovács ◽  
Géza Tóth

Abstract A planar point set of n points is called $$\gamma $$ γ -dense if the ratio of the largest and smallest distances among the points is at most $$\gamma \sqrt{n}$$ γ n . We construct a dense set of n points in the plane with $$ne^{\Omega ({\sqrt{\log n}})}$$ n e Ω ( log n ) halving lines. This improves the bound $$\Omega (n\log n)$$ Ω ( n log n ) of Edelsbrunner et al. (Discrete Comput Geom 17(3):243–255, 1997). Our construction can be generalized to higher dimensions, for any d we construct a dense point set of n points in $$\mathbb {R}^d$$ R d with $$n^{d-1}e^{\Omega ({\sqrt{\log n}})}$$ n d - 1 e Ω ( log n ) halving hyperplanes. Our lower bounds are asymptotically the same as the best known lower bounds for general point sets.


1952 ◽  
Vol 4 ◽  
pp. 424-435 ◽  
Author(s):  
I. E. Glover

V. V. Golubev, in his study [6], has constructed, by using definite integrals, various examples of analytic functions having a perfect nowhere-dense set of singular points. These functions were shown to be single-valued with a bounded imaginary part. In attempting to extend his work to the problem of constructing analytic functions having perfect, nowhere-dense singular sets under quite general conditions, he posed the following question: Given an arbitrary, perfect, nowhere-dense point-set E of positive measure in the complex plane, is it possible to construct, by passing a Jordan curve through E and by using definite integrals, an example of a single-valued analytic function, which has E as its singular set, with its imaginary part bounded.


1927 ◽  
Vol 34 (1) ◽  
pp. 36
Author(s):  
Raymond Garver

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