Let H be a collection of n hyperplanes in R(d), d greater-than-or-equa
l-to 2. For each cell c of the arrangement of H let f(i)(c) denote the
number of faces of c of dimension i, and let f(c) = SIGMA(i = 0)d-1 f
(i)(c). We prove that SIGMA(c) f(c)2 = O(n(d) log right perpendicular
d/2 left perpendicular - 1 n), where the sum extends over all cells of
the arrangement. Among other applications, we show that the total num
ber of faces bounding any m distinct cells in an arrangement of n hype
rplanes in R(d) is O(m1/2nd/2 log (right perpendicular d/2 left perpen
dicular - 1)/2n) and provide a lower bound on the maximum possible fac
e count in m distinct cells, which is close to the upper bound, and fo
r many values of m and n is OMEGA(m1/2n(d/2). (C) 1994 Academic Press,
Inc.