Let us consider the following configuration of hyperplanes in the real
vector space V with coordinates $z\_1,\ldots,z\_n$: the hyperplanes are
numbered by all the nonempty subsets $J\subset I=\{1,\ldots,n\}$, and the
hyperplane $H\_J$ is given by $\sum\_{i\in J}z\_i=0$.


Question: How many connected components does the complement
$V\setminus\cup\_{J}H\_J$ have? That is, what are they naturally numbered by?



connected components of a real hyperplane arrangement
This is not known as far as I know, and seems to be a hard problem, see the references and comments in 
[this MO question](https://mathoverflow.net/questions/62764/a-natural-refinement-of-the-a-n-arrangement-is-to-consider-all-2n-1-hyperpla).


