The Geometric Interior in Real Linear Spaces

For simplicity, we adopt the following rules: x denotes a set, r, s denote real numbers, n denotes a natural number, V denotes a real linear space, v, u, w, p denote vectors of V , A, B denote subsets of V , A1 denotes a finite subset of V , I denotes an affinely-independent subset of V , I1 denotes a finite affinelyindependent subset of V , F denotes a family of subsets of V , and L1, L2 denote linear combinations of V . Next we state four propositions: (1) Let L be a linear combination of A. Suppose L is convex and v 6= ∑ L and L(v) 6= 0. Then there exists p such that p ∈ convA \ {v} and ∑ L = L(v) · v + (1− L(v)) · p and 1 L(v) · ∑ L+ (1− 1 L(v)) · p = v. (2) Let p1, p2, w1, w2 be elements of V . Suppose that v, u ∈ conv I and u / ∈ conv I \ {p1} and u / ∈ conv I \ {p2} and w1 ∈ conv I \ {p1} and