Gale transform of a starshaped sphere

TL;DR

This paper explores the Gale transform of starshaped spheres, extending its application beyond convex polytopes to more general geometric objects, and analyzing their properties.

Contribution

It introduces the study of Gale transforms for starshaped spheres, a class not previously examined in this context, broadening the scope of Gale transform applications.

Findings

Characterization of Gale transform properties for starshaped spheres

Extension of Gale transform theory to non-convex geometric objects

Potential implications for classification of complex geometric structures

Abstract

Gale transform is a simple but powerful tool in convex geometry. In particular, the use of Gale transform is the main argument in the classification of polytopes with few vertices. Many books and documents cover the definition of Gale transform and its main properties related to convex polytopes. But it seems that there does not exist document studying the Gale transform of more general objects, such that triangulation of spheres. In this paper, we study the properties of the Gale transform of a large class of such spheres called starshaped spheres.