TITLE: Prove that 0 is in the convex hull of points chosen from each orthant
QUESTION [5 upvotes]: If we arbitrarily choose a point from each orthant in $\mathbb{R}^n$, that is we choose $2^n$ points in total, how do we prove that 0 is in the convex hull of these $2^n$ points? It seems obvious, but when I sit down and start thinking about it, I couldn't definitely find a set of $\lambda_i, i = 1,2,\cdots,2^n$ such that $0 \leq \lambda_i \leq 1, \sum_i \lambda_i = 1$ and $0 = \sum_i \lambda_i x_i$, where $x_i, i = 1,2,\cdots,2^n$ are any set of points satisfying the condition.

REPLY [3 votes]: Take in pairs the points that differ in sign for the coordinate $n$ (and only that one) and form the convex combinations such that that coordinate is cancelled.
Now you have $2^{n-1}$ points in each orthants of the diminished space.
The results holds because the convex combination of linear combinations is a convex combination.