Convex cone and conic hull

  Convex theory

Cone and convex cone

A set $C\subseteq\mathbb R^n$ is called a cone if for every $x\in C$ and $\lambda\geqslant 0$ we have $\lambda x\in C$.

A set $C$ is called a convex cone if $C$ is a cone and $C$ is convex, i.e. for any $x_1,x_2\in C$ and $\lambda_1, \lambda_2 \geqslant0$ we have

Conic combination

A conic combination of the points $x_1,\ldots, x_n$ is a point of form $\lambda_1x_1+\ldots+\lambda_nx_n$ with $\lambda_i\geqslant0\ \ \forall i=1,\ldots,n$.

The following table shows the difference between affine combination, convex combination and conic combination

Affine combinationConvex combinationConic combination
Form: $\ \lambda_1x_1 +\ldots+ \lambda_nx_n$Form: $\ \lambda_1x_1 +\ldots+ \lambda_nx_n$Form: $\ \lambda_1x_1 +\ldots+ \lambda_nx_n$
Where $\ \lambda_i\in\mathbb R$ and $\sum_{i=1}^n \lambda_i=1$Where $\ \lambda_i\geqslant0 $ and $\sum_{i=1}^n \lambda_i=1$Where $\ \lambda_i\geqslant0$

Conic hull

The set of all conic combination of points in $C$ is called the conic hull of $C$

Tip: In the definition of conic hull, we only need $\lambda_i\geqslant0$