I want to create a network where everyone can play with everyone with low enough RTT
Let's back up a little bit.
Why do you think this can't be done today? What is the specific value proposition you want to provide that is significantly different from what exists?
Using Xbox Live! as an example:
Xbox Live! lets everyone match up to everyone else -- I can play MW3 with a party of friends all over the world if I want.
The experience is slightly more laggy if we're all dispersed, and end up on a server somewhere far from me, but I can still play those people.
If you use Xbox Live! style matchmaking, but make a rule that the measured ping has to be at most X milliseconds, then you are, in effect, creating a "geographically partitioned" virtual network, that provides the latency you want. The draw-back is that you have to exclude any friends who are located far away from you, measured by ping.
So, using the existing internet, and putting servers in a few data centers, you can group players into groups with known maximum ping. If you raise the allowed ping, you can play with everyone in the world. If you lower the ping, you can play your regional area.
Now, what part of this do you want to improve?
Do you want to provide better (faster) internet service to the actual players? To do that, you need to pull fiber to each player's home. (Google is starting to do this as an experiment, btw.)
Do you want to reduce the cross-continent part of the latency? You could potentially do this by leasing your own network, but the speed improvement would be some fraction of what you'd get on the commercial internet; I doubt any consumer would see enough difference to want to pay for it.
If we take a particular game as example, maybe it would help. I can play Counter-Strike today. I can browser servers in a large part of the world. I can choose servers based on how high/low the ping is.
What part of this solution do you want to change, and how, and why?