The coax from a patio or balcony goes from the patio or balcony to inside the apartment.
I am thinking that the situation here is to get reception with a antenna on the patio or balcony and leave well enough alone.
Running coax and ground wires on the outside of the building is not likely produce favorable responds from management.
Or do you have the Ok to mount antennas on the roof or a outside wall of the building??
The Federal Antenna Law states that you have the right to install antennas on a patio or balcony that is under your direct control and is part of your apartment used by you and not the general public. In other words , patio/balcony is yours.
Here is the Federal Antenna Law.
http://www.fcc.gov/mb/facts/otard.html.
So that you are thinking clearly.
Know that the Federal Antenna Law is on -> Your Side <- and does give you rights.
_______________________________________
The following is a example.
You see those small satellite dishes on the outside of apartment buildings at locations around the city and every where you drive.
The point here is that Yes management can say yes , can install those small satellite dishes on the outside of the building that is in a location other then the direct control of the renters apartment.
This example is to point out that management can say yes to situations that are beyond what is under your direct control.
_______________________________________
The Federal Antenna Law states that patio/balcony that is under your direct control that is part of apartment unit you are renting is in fact under your direct control and you can put antenna out there.
If you look around every were you go you will see small satellite dishes and Tv Antennas on patios and balconies.