This note is an attempt to give an answer for the following old I.M.
Gelfand's question: why some important problems of integral geometry (e.g., the
Radon transform and others) are related to harmonic analysis on groups, but for
other quite similar problems such relations are not clear? In the note we
examine standard problems of integral geometry generating harmonic analysis
(the Plancherel theorem etc.) on pairs of commutative hypergroups that are in a
duality of Pontryagin's type. As a result new meaningful examples of
hypergroups are constructed.Comment: 10 pages, to be published in Doklady Mathematics, 201