Silverlight-based "area of influence" algorithm

I am in the process of developing a tool for a game called MechWarrior: Online which defines a map of an area of space called the Inner Sphere. Essentially, this map is a set of points (planets) on a Cartesian plane. Each planet can be owned by a different faction, typically represented on the map by a colour. I can colour the various planet points as different colours, but ideally what I'd like to do is show an area of influence (circle) for each faction by using something like a GeometryGroup to help with the simplification of the geometry object that using 2,240 points' circles will produce before rendering it to a Silverlight Canvas.

However, I'm not even sure that this is a good idea or the right way to approach this problem from an algorithmic standpoint, specifically because where there are overlaps of influence at the faction "borders", how do I deal with the overlaps to effectively show a reduced area of influence for both factions in the resulting geometry?