Gruber P. Convex and Discrete Geometry

8 The Brunn–Minkowski Inequality 147
by the inequality of the geometric and arithmetic mean, see Corollary 1.2. The proof
of (2) for m + n = 2 is complete. Assume now that m + n > 2 and thus in particular
m ≥ 2, say, and that (2) holds for 2,...,m+n−1 boxes. Since the boxes A1,...,Am
have pairwise disjoint interiors, there is a hyperplane HA parallel to a coordinate
there is at least one
hyperplane such that in each of the closed halfspaces H + A , H − A
of the boxes A1,...,Am. Then, among the boxes A1 ∩ H + A ,...,Am ∩ H + A , there
are 0 < m + < m proper boxes, say A +
1 ,...,A+
m + and among the boxes A1 ∩
H − A ,...,Am ∩ H − A there are 0 < m− < m proper boxes, say A −
be a hyperplane parallel to HA such that
1 ,...,A− m−.LetHB (3) V (A ∩ H + A ) = αV (A), V (A ∩ H − A ) = (1 − α)V (A),
V (B ∩ H + B ) = αV (B), V (B ∩ H − B ) = (1 − α)V (B),
where 0