Convex hull of a finite number of points in Euclidean space is bounded

Dependencies:

  1. Convex combination and convex hull
  2. Convex set

Let $X \defeq \{x^{(i)} \in \mathbb{R}^n: 1 \le i \le m\}$. Let $P$ be the convex hull of $X$. Then $|x_j|$ is finite for all $j$ and all $x \in P$.

Proof

Let $\beta \defeq \max_{i=1}^m \max_{j=1}^n |x^{(i)}_j|$. Let $\xhat \in P$. Let $\xhat = \sum_{i=1}^m \alpha_i x^{(i)}$ such that $\alpha_i \ge 0$ for all $i$ and $\sum_{i=1}^m \alpha_i = 1$. Then for any $j$, \begin{align} \|\xhat_j\| &= \left|\sum_{i=1}^m \alpha_i x^{(i)}_j\right| \\ &\le \sum_{i=1}^m |\alpha_i x^{(i)}_j| \\ &\le \sum_{i=1}^m \alpha_i \beta = \beta. \end{align}

Dependency for:

  1. Polyhedron is unbounded iff it has direction
  2. Finitely generated set is polyhedron

Info:

Transitive dependencies:

  1. Group
  2. Ring
  3. Field
  4. Vector Space
  5. Convex combination and convex hull
  6. Convex set