Actually, the GPS satellites are not geosynchronous satellites, which would imply orbits at 22,236 miles above sea level for circular orbits. From Earth, only those orbits appear stationary (in equatorial orbit) or at least varying within a relatively small area (if not in equatorial orbit).

The GPS satellites are in Medium Earth Orbit (MEO), around 12,540 miles. The GPS satellites do not hover over one area. They don't have to hover. Each satellite broadcasts its own four dimensional position in spacetime (3 space coordinates and the time, kept by accurate atomic clocks). Based on the calculated time delays from at least 4 satellites, a GPS receiver can uniquely determine the it's position.

See the Wikipedia article on GPS: https://en.wikipedia.org/wiki/Global_Positioning_System

and on geosynchronous orbit:

https://en.wikipedia.org/wiki/Geosynchronous_orbit

There are at least 3 physical arguments about where space "begins". The first was explained well by Shrike99 - a theoretical limit from von Karman that aircraft around this altitude would have to travel faster than orbital velocity to obtain enough lift to support itself. At first rounded to 100 km/62 miles in most of the world, and maybe really about 84 km/52 miles in more recent calculations. A second aerodynamic basis is a practical one. As I understand it, the USAF 50 mile definition (besides rounding down from Von Karman) is the lowest a satellite can go and still complete one orbit (because of air friction, although that would seem to depend on the shape of the satellite, so that might not be a really solid argument). A third one is not an aerodynamic argument, but from a simple observation that there is a rapid increase in atmospheric temperature below 100 km. That is suggesting that there is a qualitative boundary of sorts at that altitude, so why not call it "space" above that. It's all somewhat arbitrary anyway.

In any case, as others pointed out, all these definitions are well above the balloon height.