# Existence of a Convex Polyhedron with Respect to the Given Radii

## TL;DR

This paper proves the existence of convex configurations based on given radii in both 2D and 3D, with implications for spherical Laguerre Voronoi diagrams, advancing geometric understanding.

## Contribution

It establishes the existence of convex configurations for given radii in 2D with limited repetitions and universally in 3D, extending geometric theory.

## Key findings

- Convex configurations exist in 2D for distinct or up to four repeated radii.
- Convex configurations always exist in 3D regardless of radii repetitions.
- Implications for the existence of spherical Laguerre Voronoi diagrams.

## Abstract

Given a set of radii measured from a fixed point, the existence of a convex configuration with respect to the set of distinct radii in the two-dimensional case is proved when radii are distinct or repeated at most four points. However, we proved that there always exists a convex configuration in the three-dimensional case. In the application, we can imply the existence of the non-empty spherical Laguerre Voronoi diagram.