Complex homothetic sections and projections through a Helly type Theorem for cosets of ${\mathrm{\S }}^1$

We prove that two closed subsets of complex space ${\text{C}}^n$ with corresponding complex homothetic sections (projections) are complex homothetic. The proof uses a new Helly-type theorem for cosets of closed subgroups of ${\mathrm{\S }}^1$.