Let $x_1,\ldots,x_n$ be $n$ points in $\mathbb{R}^3$. Are there known necessary and sufficient conditions on the $x_i$'s so that the origin belongs to the convex hull of the $x_i$'s?
I did a (not so extensive) literature search, but could not find the answer yet.
Edit and comments: this problem is equivalent to whether a certain linear program has a feasible point, see orangeskid's answer below for instance. This is an interesting example of a geometric problem, for which it seems that the best one can do is to come up with an algorithm which would decide (in some reasonable time), given n points, whether or not the origin lies in the convex hull of these points or not. In some sense, this is like describing algorithmically the characteristic function of a subset we are interested in. This leads me to conjecture that there may be some subsets of R^n whose characteristic function is given as the output of an algorithm, and can not be described in any "closed form" way. I am not sure how to make this statement precise. Can someone explain to me whether what I am writing is true please?
You can do a test using linear programming as follows: for a variable affine functional $\phi$ do the linear programming $\max_{\phi} \phi(x)$ such that $\phi(x_i) \le 0 $ for all $i$ and $\phi(x)\le 1$ ( to have a bounded solution). $x$ is in the convex hull if and only if the solution to the problem is $\le 0$.