  The complex Radon transform $\hat F$ of a rapidly decreasing distribution
$F\in\mathscr{O}_C^{\prime}(\mathbb{C}^n)$ is considered. A compact set
$K\subset\mathbb{C}^n$ is called linearly convex if the set $ \mathbb{C}^n
\setminus K$ is a union of complex hyperplanes. Let $\hat K$ denote the set of
complex hyperplanes which meet $K$. The main result of the paper establishes
the conditions on a linearly convex compact $K$ under which the support theorem
for the complex Radon transform is true: from the relation $\hbox{supp}(\hat
F)\subset\hat K$ it follows that $F\in\mathscr{O}^{\prime}_C(\mathbb{C}^n)$ is
compactly supported and $\hbox{supp}(F)\subset K$.
