The discrete Pompeiu problem on the plane

Reference : The discrete Pompeiu problem on the plane


	Document type :	Scientific journals : Article
	Discipline(s) :	Physical, chemical, mathematical & earth Sciences : Mathematics
	To cite this reference:	http://hdl.handle.net/10993/33082

Title :	The discrete Pompeiu problem on the plane
Language :	English
Author, co-author :	Kiss, Gergely [University of Luxembourg > Faculty of Science, Technology and Communication (FSTC) > Mathematics Research Unit >]
	Laczkovich, Miklós [Eötvös Loránd University Budapest > Analysis]
	Vincze, Csaba [University of Debrecen]
Publication date :	Jun-2018
Journal title :	Monatshefte für Mathematik
Publisher :	Springer
Peer reviewed :	Yes (verified by ORBi^lu)
ISSN :	0026-9255
e-ISSN :	1436-5081
City :	Vienna
Country :	Austria
Keywords :	[en] 39B32 (primary) ; 30D05 (primary) ; 43A45 (secondary)
Abstract :	[en] We say that a finite subset $E$ of the Euclidean plane $\R^2$ has the discrete Pompeiu property with respect to isometries (similarities), if, whenever $f:\R^2\to \C$ is such that the sum of the values of $f$ on any congruent (similar) copy of $E$ is zero, then $f$ is identically zero. We show that every parallelogram and every quadrangle with rational coordinates has the discrete Pompeiu property with respect to isometries. We also present a family of quadrangles depending on a continuous parameter having the same property. We investigate the weighted version of the discrete Pompeiu property as well, and show that every finite linear set with commensurable distances has the weighted discrete Pompeiu property with respect to isometries, and every finite set has the weighted discrete Pompeiu property with respect to similarities.
Permalink :	http://hdl.handle.net/10993/33082

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