Is the convex hull of a set bounded? Under what conditions it's bounded?
For a given set $X \subseteq \mathbb R^n$,the convex hull of $X$ denoted by $\text{CH}(X)$ is defined as $$\text{CH}(X):=\{\sum_{j=1}^{k}\lambda_j\mathbf{x}_j:\sum_{j=1}^{k}\lambda_j=1,\lambda_1,…,\lambda_k \ge 0\}$$
Where $\mathbf{x}_j$ are vectors in $X$ and $k \in \mathbb N^+$.
If the set $X$ is bounded itself by $M$, then it's convex hull is also bounded by $M$, But is that the only way that the convex hull of a set is bounded? In other words if the convex hull of a set is bounded, then what can be said about the set itself?
Moreover I know that there are sets whose convex hull is unbounded, but can someone give me an elementary example of such sets?
Best Answer
Yes, the convex hull of a set is bounded if and only if the set is bounded. If the set is bounded, say $\lVert x\rVert\le M$ for all $x\in X$, then $\left\lVert\sum_{k=1}^s \lambda_sx_s\right\rVert\le M\sum_{k=1}^s\lvert \lambda_s\rvert=M\sum_{k=1}^s \lambda_s=M$. If $X$ is unbounded, then $\operatorname{conv}(X)\supseteq X$ is too.