I am searching for a reference, preferably with a figure, of the consensus here, which I sum up with the following quote from the cite:
"To make a clear summary of points already raised: In linear and convex optimization, where all equations and inequalities and functions are linear and convex and admissible domains are polytopes or convex sets, then the set of optimal points always contains corners of the boundary. It may contain entire facetts of the boundary, but a corner point is guaranteed."
It is almost true. If the problem is bounded, then there is always a vertex with optimal value. You can look at Theorem 10 and Corollary 11 in Karloff Linear Programming.
The Wikipedia article on linear programming gives some pictures and examples too.