The Kármán line is the altitude at which the atmospheric pressure is so thin a craft would need to achieve orbital velocity in order to maintain enough aerodynamic lift and control. Therefore, logically the Kármán line itself should belong to outer space already, according to that definition. However, this document states that you have to go beyond 100 km to be recognized an astronaut by the FAI. While the actual Kármán line is lower anyway, logically you should be recognized a space traveller if you reached an apogee of 100,000 meters. But if I understand the document correctly, for the FAI you wouldn't; you have to reach e.g. 100,001 meters to be recognized an astronaut by the FAI.
I believe the U.S. space definition is that you have to go above 50 miles (or 80,467.2 meters) to be recognized an astronaut but that's logical since it may be tied to a certain atmospheric pressure. But if you set the space border according to the Kármán line, the line itself should belong to space, or did I miss something?
Questions like this might play and have played a role in flights of spaceplanes. Mike Melvill reached an apogee of 100,124 meters in SpaceShipOne flight 15P. That's the lowest FAI-recognized spaceflight.