Question

Let $S \subset \mathbb{R}^k$ be a set. Show that the convex hull of , denoted by , is equal to the set $\bigcup \left\{ (1-t)x + tx \, | \, t \in [0,1], x,y \in S \right\}$

cvx(S)

Answer 1

Convex hull of a set S is the set of all convex linear combinations of elements in S. Let $x,y \in S ; t \in [0,1]$ then $(1-t)x+ty \in \cup_t \left \{ (1-t)x+ty| t \in [0,1],x,y \in S \right \}$ which is a convex linear combination of x and y hence it is an element in cvx(S). Hence every element of the given union is an element of the convex hull. Therefore $\cup_t \left \{ (1-t)x+ty| t \in [0,1],x,y \in S \right \} \subseteq cvx(S)$ ----(1)

Now let x is an element in S then $x = 1.x | 1 \in [0,1] \Rightarrow x \in \cup_t \left \{ (1-t)x+ty| t \in [0,1],x,y \in S \right \}$ , therefore$S\subseteq \cup_t \left \{ (1-t)x+ty| t \in [0,1],x,y \in S \right \}$. Now the union contains all possible convex linear combination of elements in S and cvx(S) is the smallest set containing all such elements thus we have $cvx(S)\subseteq \cup_t \left \{ (1-t)x+ty| t \in [0,1],x,y \in S \right \}$ ----(2).

From (1) and (2) we get $cvx(S)=\cup_t \left \{ (1-t)x+ty| t \in [0,1],x,y \in S \right \}$ .