Operators preserve convexity

  Convex theory

In order to determine or establish convexity of sets, it is useful to review some operators that preserve the convexity.


Firstly, convexity is preserved under intersection. Namely, if $C_1$ and $C_2$ are convex then $C_1\cap C_2$ is convex.

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 $\lambda_i\geqslant0$