Corollary 1

The convex envelope of a subset A of E is identical with the set of linear combinations $\sum_i \lambda_i x_i$, where $(x_i)$ is any finite family of points of A, the numbers $\lambda_i > 0, \text{ for all } i$ and $\sum_i \lambda_i = 1$.

The dimension of the affine linear variety (A, II, § 9.3) generated by the convex set A is called the dimension of A.

Let E be a vector space. The convex envelope C, of the balanced envelope of a set A in E is called the balanced convex envelope (or the symmetric convex envelope) of A; clearly it is the smallest symmetric convex set that contains A; it is also the convex envelope of $A \cup (-A)$, since every point of the balanced envelope of A belongs to a segment with extremities a and −a where $a \in A$. The set C coincides with the set of linear combinations $\sum_i \lambda_i x_i$ where $x_i \in A$ and $\sum_i |\lambda_i| \leq 1$; for it is clear that this set of points is convex and contains A and −A; it is sufficient to prove that it is contained in C, and for this we need consider only those linear combinations for which $\mu = \sum_i |\lambda_i| > 0$; we can then write $\sum_i \lambda_i x_i = \mu \cdot \sum_i \alpha_i y_i$ with $\alpha_i = \lambda_i/\mu$ and $y_i = x_i$, if $\lambda_i \geq 0$; and $\alpha_i = -\lambda_i/\mu$; $y_i = -x_i$ if $\lambda_i < 0$; clearly $\sum_i \alpha_i = 1$, and our assertion is proved.