convex set

Convex set

A set $C$ is convex if the line segment between any two points in $C$ lies in $C$, i.e for any $x_1,x_2\in C$ and any $\theta \in [0,1]$, we have

$$\theta x_1+(1-\theta )x_2\in C$$

In other words: It is closed under convex combination of its points.

The intersecttion of two convex sets is a convex set

Proof: Let $C_1,C_2$ be convex sets, and their intersection be $C=C_1\cap C_2$.
if $x,y\in C$, then $x,y\in C_1$ and $x,y\in C_2$.
Both are convex, thus

$$\theta x+(1-\theta )y\in C_1\cap C_2=C$$

for all $\theta \in [0,1]$ for all $x,y\in C$