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On a forgotten conjecture from a famous paper of Erdös

Content Provider	ACM Digital Library
Author	Bárány, Imre Roldán-Pensado, Edgardo
Abstract	In his paper "On sets of distances of n points", Paul Erdos conjectured that every convex curve contains a point P such that every circle centered at P intersects the curve in at most 2 points. This conjecture is false: If T is an equilateral triangle with boundary T, for any point P on T there is a circle centered at P that intersects T at 4 points. But perhaps the number 2 in Erdos's conjecture can be replaced by some other number. Given a convex body K in the plane, let N(K) be the smallest number for which there is a point P in K such that every circle centered at P intersects K in at most N(K) points. Erdos's original conjecture states that N(K)≤ 2 for every convex body K. We give an example of a convex body K for which N(K)=6 and we show that N(K) is finite for every convex body K. We believe that N(K) is bounded by some constant, probably by 6, but so far we have not been able to find any global upper bound. Part of the difficulty may come from the following. Given a number \$n\$, let J(K,n) be the set of points P in K such that there is a circle centered at P that intersects K in at least n points. Then N(K) is the largest N such that J(K,N)=bd(K). For every ε>0, there is a convex body K such that \|J(K,∞)\|\|bd(K)\|>1-ε, where \|X\| represents the 1-dimensional Hausdorff measure of X. In the Baire category sense, for most convex bodies K the set J(K,n) contains most points of K.
Ending Page	80
Page Count	4
Starting Page	77
File Format	PDF
ISBN	9781450320313
DOI	10.1145/2462356.2462397
Language	English
Publisher	Association for Computing Machinery (ACM)
Publisher Date	2013-06-17
Publisher Place	New York
Access Restriction	Subscribed
Subject Keyword	Erdos problem Geometry Convexity
Content Type	Text
Resource Type	Article

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