The convex hull of a set of points in S is the boundary of the smallest convex region that contain all the points of S inside it or on its boundary.

OR

Let $Ssubseteq mathbb{R}^n$ The convex hull of S, denoted $Coleft ( S right )$ by is the collection of all convex combination of S, i.e., $x in Coleft ( S right )$ if and only if $x in displaystylesumlimits_{i=1}^n lambda_ix_i$, where $displaystylesumlimits_{1}^n lambda_i=1$ and $lambda_i geq 0 forall x_i in S$

Remark − Conves hull of a set of points in S in the plane defines a convex polygon and the points of S on the boundary of the polygon defines the vertices of the polygon.

Theorem $Coleft ( S right )= left { x:x=displaystylesumlimits_{i=1}^n lambda_ix_i,x_i in S, displaystylesumlimits_{i=1}^n lambda_i=1,lambda_i geq 0 right }$ Show that a convex hull is a convex set.

Proof

Let $x_1,x_2 in Coleft ( S right )$, then $x_1=displaystylesumlimits_{i=1}^n lambda_ix_i$ and $x_2=displaystylesumlimits_{i=1}^n lambda_gamma x_i$ where $displaystylesumlimits_{i=1}^n lambda_i=1, lambda_igeq 0$ and $displaystylesumlimits_{i=1}^n gamma_i=1,gamma_igeq0$

For $theta in left ( 0,1 right ),theta x_1+left ( 1-theta right )x_2=theta displaystylesumlimits_{i=1}^n lambda_ix_i+left ( 1-theta right )displaystylesumlimits_{i=1}^n gamma_ix_i$

$theta x_1+left ( 1-theta right )x_2=displaystylesumlimits_{i=1}^n lambda_i theta x_i+displaystylesumlimits_{i=1}^n gamma_ileft ( 1-theta right )x_i$

$theta x_1+left ( 1-theta right )x_2=displaystylesumlimits_{i=1}^nleft [ lambda_itheta +gamma_ileft ( 1-theta right ) right ]x_i$

Considering the coefficients,

$displaystylesumlimits_{i=1}^nleft [ lambda_itheta +gamma_ileft ( 1-theta right ) right ]=theta displaystylesumlimits_{i=1}^n lambda_i+left ( 1-theta right )displaystylesumlimits_{i=1}^ngamma_i=theta +left ( 1-theta right )=1$

Hence, $theta x_1+left ( 1-theta right )x_2 in Coleft ( S right )$

Thus, a convex hull is a convex set.