This should work: -Create from all the edges of your polyhedron Rays (make sure that the rays are not infinite, but have the length of the tested edge), -Intersect them with all triangles of polyhedron.
"multiplayer real-time strategy game" -> Sounds like allot of raw processing power. Meaning java simply wont cut it. I suggest c++ or C# at least.
c++ is a very powerful programming language. It will take some time to understand it properly and a very long time to master it (if possible).
I suggest to learn c++. Since its a base for game development in my opinion. if you can code C++, other languages will be fairly easy to grasp.
If you want to have a faster learning curve, try C# (XNA). But i still recommend C++ over it, but its partially because im a C++ junky.