On page 2, you say "In linear programming, the maximum of an objective function tends to occur on a vertex of the space." Here "tends to" seems unnecessary hedging -- I think this is just a theorem!

It is. If there exists an optimal solution, at least one vertex will be optimal, and as RyanCarey points out, if a hyperface is optimal it will have at least one vertex.

A stronger statement is that the Simplex algorithm will always return an optimal vertex (interior point algorithms will return the center of the hyperface, which is only a vertex if that's the only optimal point).