COUNTINGTHE NUMBEROF BOUNDED DOMAINSSEPARATEDBY HYPERPLANES

When some hyperplanes Hi,... , Hmof the n-dimensional Euclidean space Rn are given in general position, Schlafli has determined the number of the bounded connected components in Rw- UT=iHi, the complementary set of the union of the hyperplanes. It is equal to the binomial coefficient (^w~ J ,which is alsoequaltothe numberofvertices which are the intersections of nhyperplanes in Hi,... , Hm-i. Although Sch&fli's proof is implicit and intuitive, the fact reflects an interesting aspect concerning configurations of hyperplanes. We clarify how the condition of general position works, and re-prove the fact in all of its details.