Abstract
We say that a finite subset E of the Euclidean plane \(\mathbb {R}^2\) has the discrete Pompeiu property with respect to isometries (similarities), if, whenever \(f:\mathbb {R}^2\rightarrow \mathbb {C}\) 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.
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Communicated by A. Constantin.
G. Kiss is partially supported by the project R-AGR-0500-MRO3 of the University of Luxembourg, and also by the Hungarian National Research, Development and Innovation Office, Grant No. NKFIH 104178.
M. Laczkovich is partially supported by the Hungarian National Research, Development and Innovation Office, Grant No. NKFIH 104178.
Cs. Vincze is supported by the University of Debrecen’s internal research project RH/885/2013.
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Kiss, G., Laczkovich, M. & Vincze, C. The discrete Pompeiu problem on the plane. Monatsh Math 186, 299–314 (2018). https://doi.org/10.1007/s00605-017-1136-9
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DOI: https://doi.org/10.1007/s00605-017-1136-9