Exercise 8, § 2

Let $(A_i)_{1 \leq i \leq p}$ be a finite family of convex subsets of a vector space E; let $W_i$ be the subspace obtained by a translation from the affine linear variety generated by $A_i$ ($1 \leq i \leq p$). If $W = \sum_{i=1}^p W_i$, show that the affine linear variety generated by the convex set $\sum_{i=1}^p \lambda_i A_i$ (where $\lambda_i$ are non-zero numbers) is obtained from W by a translation.