Smooth maps, null-sets for integralgeometric measure and analytic capacity
type: Item
instance of: scholarly article
title: Smooth maps, null-sets for integralgeometric measure and analytic capacity
zbMATH Open document ID: 0589.28006
DOI: 10.2307/1971273
author: Pertti Mattila
published in: Annals of Mathematics. Second Series
publication date: 1986-01-01T00:00:00Z
review text: A Borel subset E of the plane \(R^ 2\) has zero integralgeometric measure, \(I^ 1(E)=0\), if the orthogonal projections of E on almost all lines through the origin are of length zero. It is shown that a \(C^ 2\) diffeomorphism between two open subsets of \(R^ 2\) preserves sets of integralgeometric measure zero if and only if it maps every line segment onto a line segment. In particular, the condition \(I^ 1(E)=0\) is not conformally invariant. This disproves Vitushkin's conjecture according to which the compact null-sets for integralgeometric measure would be the same as the compact null-sets for analytic capacity.
Mathematics Subject Classification ID: 28A75
Mathematics Subject Classification ID: 30C85
zbMATH DE Number: 3946510
zbMATH Keywords: integralgeometric measure
zbMATH Keywords: orthogonal projections
zbMATH Keywords: \(C^ 2\) diffeomorphism
zbMATH Keywords: Vitushkin's conjecture
zbMATH Keywords: analytic capacity
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.2307/1971273
OpenAlex ID: W2325072092
Wikidata QID: Q56813982
description: scientific article; zbMATH DE number 3946510