Some Properties for Convex Combinations

This is a continuation of [6]. In this article, we proved that convex combination on convex family is convex. and [9] provide the notation and terminology for this paper. 1. CONVEX COMBINATIONS ON CONVEX FAMILY The following propositions are true: (1) For every non empty RLS structure V and for all convex subsets M, N of V holds M ∩ N is convex. (2) Let V be a real unitary space-like non empty unitary space structure, M be a subset of V , F be a finite sequence of elements of the carrier of V , and B be a finite sequence of elements of R. Suppose M = {u; u ranges over vectors of V : i : natural number (i ∈ dom F ∩ dom B ⇒ v : vector of V (v = F(i) ∧ (u|v) ≤ B(i)))}. Then M is convex. (3) Let V be a real unitary space-like non empty unitary space structure, M be a subset of V , F be a finite sequence of elements of the carrier of V , and B be a finite sequence of elements of R. Suppose M = {u; u ranges over vectors of V : i : natural number (i ∈ dom F ∩ dom B ⇒ v : vector of V (v = F(i) ∧ (u|v) < B(i)))}. Then M is convex. (4) Let V be a real unitary space-like non empty unitary space structure, M be a subset of V , F be a finite sequence of elements of the carrier of V , and B be a finite sequence of elements of R. Suppose M = {u; u ranges over vectors of V : i : natural number (i ∈ dom F ∩ dom B ⇒ v : vector of V (v = F(i) ∧ (u|v) ≥ B(i)))}. Then M is convex. (5) Let V be a real unitary space-like non empty unitary space structure, M be a subset of V , F be a finite sequence of elements of the carrier of V , and B be a finite sequence of elements of R. Suppose M = {u; u ranges over vectors of V : i : natural number (i ∈ dom F ∩ dom B ⇒ v : vector of V (v = F(i) ∧ (u|v) > B(i)))}. Then M is convex. (6) Let V be a real linear space and …