Starting from some distribution of n points on a sphere (where n>3), if we take the Voronoi diagram on the sphere of those points, the resulting regions could have various numbers of sides.
If we allow sides to be split into multiple sides by introducing new vertices along their length, can we always make it so that each region has exactly 6 sides?
(Assuming none of the original Voronoi regions have more than 6 sides to begin with)
An illustration to hopefully make this clearer - splitting the edges at the black points results in all regions having 6 sides:
Is this equivalent to any other solved question?
Follow up - If we drop the restriction that the original Voronoi regions don't have more than 6 sides, and also allow edges to be 'collapsed' (by bringing their end points together) can we still always get to 6 sided regions while also guaranteeing the regions all stay convex, and each contains their original point?
This is not always possible; consider placing the points at the centers of the faces of a soccer ball. Then there are $12$ pentagonal faces surrounded by hexagonal faces.