Support theorem for the complex Radon transform (Q1387267)

In its real version the theorem about the support of the Radon transform asserts that a compactly supported function with vanishing integral over all hyperplanes disjoint from a compact convex \(K\) in \(\mathbb{R}^n\) vanishes off \(K\). A counterexample to the corresponding version of the theorem in \(\mathbb{C}^n\) was provided by the author [Mat. Zametki 54, No. 3, 152-154 (1993; Zbl 0852.44002)]. The present paper provides two positive results clarifying the situation.