An optimal extension of Marstrand's plane-packing theorem
type: Item
instance of: scholarly article
title: An optimal extension of Marstrand's plane-packing theorem
zbMATH Open document ID: 1060.28001
published in: Archiv der Mathematik
publication date: 2004-03-07T00:00:00Z
review text: The aim of this paper is to prove an extension of Marstrand's plane packing theorem [\textit{J. Marstrand}, ``Packing planes in \(R^3\)'', Mathematika 26, 180--183 (1979; Zbl 0413.28008)], namely, it is shown that if \(F\) is a subset of the 2--dimensional unit sphere in \(\mathbb R^{3}\) having Hausdorff dimension strictly greater than 1, and \(E\subset \mathbb{R}^{3}\) is a Borel set such that for each \(e\in F\), \(E\) contains a plane perpendicular to the vector \(e\), then \(E\) has positive \(3\)--dimensional Lebesgue measure. This result is optimal in the sense that the condition on the Hausdorff dimension cannot be relaxed.
reviewed by: Emilia Petrisor
Mathematics Subject Classification ID: 28A75
Mathematics Subject Classification ID: 28A78
Mathematics Subject Classification ID: 28A80
zbMATH DE Number: 2051595
zbMATH Keywords: Hausdorff dimension
zbMATH Keywords: plane packing theorem
author: Themis Mitsis
MaRDI profile type: Publication
full work available at URL: s00013-003-4705-1
OpenAlex ID: W2068334918
DOI: 10.1007/S00013-003-4705-1
description: scientific article; zbMATH DE number 2051595