RTS/4x area of influence (or combining multiple circles into one shape?)

[wind axle]

How would you achieve an effect like this?

1. The 'sources of influence' can be placed at any Vector2 position
2. Each 'source of influence' can have a different 'influence power' so one can give more radius than the other
3. You have to be able to tell when a given point is inside a player's 'influence area'
4. You have to figure out a "mid point" between different player's 'areas of influence' when they overlap
5. You have to visualize the area of influence (let's stick to 2d)
6. It can't utterly obliterate the performance when there's 500+ 'sources of influence'

[surreal herald]

This is basically a smoothed, weighted voronoi graph