Convex set

There is a homology between a line segment and a convex set. It is helpful to understand the convex set. A line, a line segment, and one sideline has homology to an affine set, a convex set, and a cone. A line is $\{y|y={\theta }_1{x}_1+{\theta }_2{x}_2,{\theta }_1+{\theta }_2=1\}$ if ${\theta }_1,{\theta }_2\in R$, a line segment is if ${\theta }_1,{\theta }_2>0$ and an one side line if any ${\theta }_1,{\theta }_2<0$.

A set $C$ is affine set if $yC$ and $\{y|y={\theta }_1{x}_1+{\theta }_2{x}_2,{\theta }_1+{\theta }_2=1,x_1,x_2\in C,{\theta }_1,{\theta }_2\in R\}$. a convex set is if ${\theta }_1,{\theta }_2>0$ and a cone if any ${\theta }_1,{\theta }_2<0$.

An affine set is a convex set. But all convex set is not an affine set. It looks the convex set has a stronger condition than affine set i.e. positivity of $\theta$. But in fact, the convex set has a stronger condition on what it should contain. Because an affine set contains more than a convex set, an affine set satisfies the condition to be a convex set.