Geodesic
Sets with the Pompeiu Property on the Plane and on the Sphere
Matematičeskie zametki, Tome 87 (2010) no. 1, pp. 69-82

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We obtain new sufficient conditions under which a set on the plane has the Pompeiu property. This result allows us to construct first examples of domains with the Pompeiu property with non-Lipschitz (and even fractal) boundary. In addition, we study the problem of determining the least radius of the ball on the sphere in which a given set is a Pompeiu set. We obtain the solution of this problem in the case of a biangle and a spherical half-disk. We also consider some applications to questions of complex analysis.
Mots-clés : Pompeiu problem, biangle
Keywords: Pompeiu property, non-Lipschitz boundary, spherical half-disk, Koch snowflake, Morera-type theorems, Laplace operator. 
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     author = {V. V. Volchkov and Vit. V. Volchkov},
     title = {Sets with the {Pompeiu} {Property} on the {Plane} and on the {Sphere}},
     journal = {Matemati\v{c}eskie zametki},
     pages = {69--82},
     publisher = {mathdoc},
     volume = {87},
     number = {1},
     year = {2010},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_2010_87_1_a7/}
}
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V. V. Volchkov; Vit. V. Volchkov. Sets with the Pompeiu Property on the Plane and on the Sphere. Matematičeskie zametki, Tome 87 (2010) no. 1, pp. 69-82. http://geodesic.mathdoc.fr/item/MZM_2010_87_1_a7/