Game of area

I thought of this game, where you both start off controlling a half of the board like so:

And the players take turns picking points on this grid, and any point on the grid that is closer to one of the red player's points than any of the blue player's points gets colored red, and vice versa. Any point that is equally close is grey and doesn't count for either player.

So here might be the first few moves of a game on an 800 by 800 point board:
Red(133,133)
Blue(470,200)
Red(500,160)
Blue(750,100)
Red(50, 200)
Blue(744, 50)
Red(150, 200)

After those moves the board would look like:

I chose those moves kind of randomly but I think you get the idea.