[Math] Is convex hull of a finite set of points in $\mathbb R^2$ closed

Question

Is the convex hull of a finite set of points in $\mathbb R^2$ closed? Intuitively, yes. But not sure how to show that. Thanks!

Answer 1

Generally, the convex hull $\{\lambda_1x_1+\dots+\lambda_nx_n\colon0\le\lambda_1,\dots,\lambda_n\le1,\lambda_1+\dotsb+\lambda_n=1\}$ of $x_1,\dotsc,x_n\in\mathbb R^d$ is compact, hence closed, since it's the continuous image of the closed simplex $\{(\lambda_1,\dotsc,\lambda_n)\colon0\le\lambda_1,\dots,\lambda_n\le1,\lambda_1+\dotsb+\lambda_n=1\}\subseteq\mathbb R^n$.

To fill in the details, you can prove by induction that the closed simplex is really compact as Hamcke mentioned in the comment, but you can do it simpler: as a subspace of $\mathbb R^n$, it's obviously bounded and it could be written as an intersection of a (in fact finite) collection of closed subsets of $\mathbb R^n$.