Exercise 9, § 2

Let A be a subset of the space $\mathbf{R}^n$. a) Show that the convex envelope of A is identical with the set of points $\sum_{i=0}^n \lambda_i x_i$, where $x_i \in A$, $\lambda_i \geq 0$ for $0 \leq i < n$, and $\sum_{i=0}^n \lambda_i = 1$. (Establish the following lemma; if $p + 1$ points $x_i$ ($0 \leq i \leq p$) form an affinely dependent system (that is to say there exists a relation $\sum_{i=0}^p \beta_i x_i = 0$ where the $\beta_i$ are not all zero and $\sum_{i=0}^p \beta_i = 0$) and if $x = \sum_{i=0}^p \alpha_i x_i$, where the $\alpha_i$ are $\geq 0$ and $\sum_{i=0}^p \alpha_i = 1$, then there exists an index $k \leq p$ and $p$ numbers $\gamma_i$ ($0 \leq i \leq p, i \neq k$) such that $\gamma_i \geq 0$ for all $i$, $\sum_{i \neq k} \gamma_i = 1$ and $x = \sum_{i \neq k} \gamma_i x_i$; for this compare those of the $\alpha_i / \beta_i$ that are defined.) b) Let $a$ be a point of the convex envelope of A which does not belong to the convex envelope of any subset of A with at most $n$ points. Show then that A contains at least $n + 1$ connected components. (We can suppose that $a = 0$; let $(b_i)_{0 \leq i \leq n}$ be a family of $n + 1$ affinely independent points of A such that 0 belongs to the convex envelope of the $b_i$ (cf. a)). For each index $i$, let $C_i$ be the pointed convex cone of vertex 0 generated by the $b_j$ with indices $j \neq i$; show that A does not meet the frontier of any of the cones — $C_i$. c) If C is a pointed cone with vertex 0 in $\mathbf{R}^n$, show that the convex envelope of C is the set of points $\sum_{i=1}^n x_i$, where $x_i \in C$ for $1 \leq i \leq n$.