About the number of connected components in arrangements of hyperplanes

Abstract: We consider arrangements of n hyperplanes of codimension one in a real projective space of dimension d. Let us denote by F the maximal possible number f of connected components of the complement in the projective space of dimension d to the union of n hyperplanes. We prove that for sufficiently large n and for d>3 almost all integers between n and F could be realized as the numbers of regions for some arrangements of n hyperplanes in the projective space. This fact was known before for d=2,3.