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Dantzig later published his "homework" as a thesis to earn his doctorate.The column geometry used in this thesis gave Dantzig insight that made him believe that the Simplex method would be very efficient.The algorithm always terminates because the number of vertices in the polytope is finite; moreover since we jump between vertices always in the same direction (that of the objective function), we hope that the number of vertices visited will be small.
It can also be shown that, if an extreme point is not a maximum point of the objective function, then there is an edge containing the point so that the objective function is strictly increasing on the edge moving away from the point.
If the edge is finite, then the edge connects to another extreme point where the objective function has a greater value, otherwise the objective function is unbounded above on the edge and the linear program has no solution.
In geometric terms, the feasible region defined by all values of is a (possibly unbounded) convex polytope.
An extreme point or vertex of this polytope is known as basic feasible solution (BFS).
Dantzig's core insight was to realize that most such ground rules can be translated into a linear objective function that needs to be maximized.