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

TL;DR

This paper establishes a Helly-type theorem for cosets of closed subgroups of the circle group and applies it to prove that two closed subsets of complex space with homothetic sections or projections are themselves homothetic.

Contribution

It introduces a new Helly-type theorem for cosets of closed subgroups of  and uses it to characterize complex homothety of subsets in ^n.

Findings

Proves a Helly-type theorem for cosets of .

Shows that homothetic sections imply homothety of the entire sets.

Provides a new tool for analyzing complex homothety in ^n.

Abstract

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