From spherical to Euclidean illumination

TL;DR

This paper explores the illumination problem for convex bodies in spherical spaces, providing solutions for large subclasses and deriving implications for Euclidean spaces, including special Koebe polyhedra in three dimensions.

Contribution

It introduces the spherical illumination problem and connects it to Euclidean cases, offering new solutions for large classes of convex bodies.

Findings

Solved illumination problem for a large subfamily of convex bodies in spherical spaces.

Derived a combinatorial version of the classical Euclidean illumination problem.

Applied results to special Koebe polyhedra in three dimensions.

Abstract

In this note we introduce the problem of illumination of convex bodies in spherical spaces and solve it for a large subfamily of convex bodies. We derive from it a combinatorial version of the classical illumination problem for convex bodies in Euclidean spaces as well as a solution to that for a large subfamily of convex bodies, which in dimension three leads to special Koebe polyhedra.