Here are some recommended readings:
Keywords: linear and affine independence, convexity, Helly's theorem, Charathéodory's theorem, convex polytopes and their faces, duality of polytopes.
Keywords: convex polytopes and face numbers, f-vector, McMullen's upper bound theorem and proof, simplex method.
Those more computationally oriented may want
to look at some additional materials (for Fukuda's part):