Problem: Let $S \subset \mathbb R^n$, where $S=\{x_1,x_2,...x_{2n}\}$, $x_i \in B_r(e_i)$ and $x_{n+i} \in B_r(-e_i)$, provided that $1 \leq i \leq n$, $0 < r<n^{-4}$, and $\{e_1,...,e_n\}$ is the standard basis of $\mathbb R^n$. Does $0\in C(S)$, the convex hull of $S$?
Solution: Consider the convex combination of the points of $S$, given by $\sum_{k=1}^{2n} \lambda _kx_k$, where $\lambda _k \in [0,1], \sum_{k=1}^{2n} \lambda _k=1$, and $x_j=\sum_{i=1}^{n} a_{ji}e_i, 1 \leq j \leq 2n$, provided that $x_i \in B_r(e_i)$ and $x_{n+i} \in B_r(-e_i)$.
Now, $\sum_{k=1}^{2n} \lambda _kx_k=\sum_{k=1}^{2n} \sum_{i=1}^{n} \lambda _k a_{ki}e_i = \sum_{k=1}^{2n} \lambda _k (\sum_{i=1}^{n} a_{ki}e_i) = \sum_{i=1}^{n}(\sum_{k=1}^{2n} \lambda _k a_{ki})e_i$
I am stuck at this point. How do I proceed further from here to answer the required?