Positive analytic capacity but zero Buffon needle probability
type: Item
instance of: scholarly article
title: Positive analytic capacity but zero Buffon needle probability
zbMATH Open document ID: 0653.30016
author: Peter Jones
author: Takafumi Murai
published in: Pacific Journal of Mathematics
publication date: 1988-01-01T00:00:00Z
review text: One of Vitushkin's famous questions is whether the analytic capacity \(\gamma\) (\(\cdot)\) and the Buffon length Bu(\(\cdot)\) have the same null sets. (The Buffon length of a planar set E is the average length of the orthogonal projection of E (averaging over all possible directions)). \textit{P. Mattila} proved that the answer to this question is no [Ann. Math., II. Ser. 123, 303-309 (1986; Zbl 0589.28006)]. However, Mattila's argument did not give which of the implications \[ 1)\quad \gamma (E)=0\quad \Rightarrow \quad Bu(E)=0,\quad 2)\quad Bu(E)=0\quad \Rightarrow \quad \gamma (E)=0 \] are false. In this paper an explicit construction is given of a set \(E_ 0\) such that \(\gamma (E_ 0)=1\) and \(Bu(E_ 0)=0\). \(E_ 0\) is a ``crank'', i.e. a limit of certain families of line segments parallel to the x-axis.
Mathematics Subject Classification ID: 30C85
Mathematics Subject Classification ID: 28A75
zbMATH DE Number: 4065373
zbMATH Keywords: analytic capacity
zbMATH Keywords: Buffon length
zbMATH Keywords: null sets
zbMATH Keywords: crank
reviewed by: B. Øksendal
MaRDI profile type: Publication
full work available at URL: pjm.1988.133.99
OpenAlex ID: W2066811174
DOI: 10.2140/PJM.1988.133.99
description: scientific article; zbMATH DE number 4065373