A set with finite curvature and projections of zero length
type: Item
instance of: scholarly article
title: A set with finite curvature and projections of zero length
zbMATH Open document ID: 0973.30022
author: Helen Joyce
author: Peter Mörters
published in: Journal of Mathematical Analysis and Applications
publication date: 2001-11-25T00:00:00Z
full work available at URL: http://nbn-resolving.de/urn/resolver.pl
review text: A compact set \(E\) in the plane has finite Melnikov (Menger) curvature if it supports a measure \(\mu\) such that \(\mu(\Delta)\leq |\Delta|\) for a very closed disk \(\Delta\) of diameter \(|\Delta|\) and such that \[ \int_E\int_E\int_E R(x,y,z)^{-2} d\mu(x) d\mu(y) d\mu(z)<\infty, \] where \(R(x,y,z)\) is the radius of the circle that passes through the points \(x,y,z\). Recently \textit{G. David} and \textit{J. C. Léger} [Ann. Math. 149, 831-869 (1999)] have proved that a compact planar set of finite length has finite Melnikov curvature if and only if it projects onto a set of positive length in almost all directions. The authors of the paper under review present an example showing that this theorem fails without the finite length assumption. The construction is similar to one by \textit{M. A. Martin} and \textit{P. Mattila} [Trans. Am. Math. Soc. 305, 293-315 (1988; Zbl 0643.28009)]. The Melnikov curvature has played an important role in the old open problem of characterizing the removable sets for bounded analytic functions. The authors remark that their example shows that the finite length assumption is necessary in Vitushkin's conjecture [recently proved by \textit{G. David}, Rev. Mat. Iberoam. 14, No. 2, 369-479 (1998; Zbl 0913.30012)]. This fact was known because of a more complicated example by \textit{P. W. Jones} and \textit{T. Murai} [Pac. J. Math. 133, No. 1, 99-114 (1988; Zbl 0653.30016)].
reviewed by: Dimitrios Betsakos
Mathematics Subject Classification ID: 30C85
zbMATH DE Number: 1497292
zbMATH Keywords: Melnikov (Menger) curvature
zbMATH Keywords: rectifiability
zbMATH Keywords: Hausdorff measure
zbMATH Keywords: analytic capacity
MaRDI profile type: Publication
OpenAlex ID: W1967523109
cites work: Unrectifiable 1-sets have vanishing analytic capacity
cites work: Positive analytic capacity but zero Buffon needle probability
cites work: Menger curvature and rectifiability
cites work: k-Dimensional Regularity Classifications for s-Fractals
cites work: Q4328334
cites work: Smooth maps, null-sets for integralgeometric measure and analytic capacity
cites work: Singular integrals, analytic capacity and rectifiability
cites work: The Cauchy integral, analytic capacity, and uniform rectifiability
cites work: Analytic capacity: discrete approach and curvature of measure
DOI: 10.1006/JMAA.2000.6831
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description: scientific article; zbMATH DE number 1497292