Generalizations of the shadow problem

TL;DR

This paper extends the shadow problem to n-dimensional Euclidean and complex spaces, providing conditions for constructing shadows using convex sets, translations, and homotheties, with specific results on balls on spheres.

Contribution

It generalizes the shadow problem to higher-dimensional Euclidean and complex spaces, offering new geometric constructions and conditions for shadows.

Findings

Solution for the shadow problem in n-dimensional Euclidean space.

Identification of sufficient balls on spheres for shadows in complex space.

Extension of shadow problem solutions to hypercomplex spaces.

Abstract

We received a solution of the shadow problem in n-dimensional Euclidean space for a family of sets, constructing from any convex domain having nonempty interior with the help of parallel translations and homotheties. We find a number of balls with centers on the sphere, sufficient for giving a shadow in n-dimensional complex (hypercomplex) space.