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 ORBilu) 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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