Brouwer Fixed Point Theorem in the General Case

For simplicity, we adopt the following convention: n is a natural number, p, q, u, w are points of En T, S is a subset of En T, A, B are convex subsets of En T, and r is a real number. Next we state several propositions: (1) (1− r) · p+ r · q = p+ r · (q − p). (2) If u, w ∈ halfline(p, q) and |u− p| = |w − p|, then u = w. (3) Let given S. Suppose p ∈ S and p 6= q and S ∩ halfline(p, q) is Bounded. Then there exists w such that (i) w ∈ FrS ∩ halfline(p, q), (ii) for every u such that u ∈ S ∩ halfline(p, q) holds |p− u| ≤ |p−w|, and (iii) for every r such that r > 0 there exists u such that u ∈ S∩halfline(p, q) and |w − u| < r.