Exercise 10, § 2

Let C be the convex envelope of a subset A of $\mathbf{R}^n$, and let $a$ be an interior point of C. Show that there exist $2n$ points $x_i \in A$ ($1 \leq i \leq 2n$) such that $a$ is interior to $C_0$, the convex envelope of the $x_i$. (Suppose $a = 0$, and argue by induction on $n$, noting that by exerc. 9a) there exists a set of $k + 1$ points $y_j$ of A ($0 \leq j \leq k, 1 \leq k \leq n$), affinely independent and such that, if V is the affine linear variety generated by the $y_j$, then $0 \in V$ and, relative to V, 0 is interior to the convex envelope of the set of the $y_j$. Then project C on E/V and show that 0 is interior to this projection relative to E/V). Show that in the above statement $2n$ cannot be replaced by $2n - 1$.