On the Equality Conditions of the Brunn-minkowski Theorem

This article describes a new proof of the equality condition for the Brunn-Minkowski inequality. The Brunn-Minkowski Theorem asserts that, for compact convex sets K,L ⊆ Rn, the n-th root of the Euclidean volume Vn is concave with respect to Minkowski combinations; that is, for λ ∈ [0, 1], Vn((1− λ)K + λL) ≥ (1− λ)Vn(K) + λVn(L). The equality condition asserts that if K and L both have positive volume, then equality holds for some λ ∈ (0, 1) if and only if K and L are homothetic. Denote n-dimensional Euclidean space by R. Given compact convex subsets K,L ⊆ R and a, b ≥ 0, denote aK + bL = {ax+ by | x ∈ K and y ∈ L}. An expression of this form is called a Minkowski combination or Minkowski sum. Since K and L are convex sets, the set aK + bL is also convex. Convexity also implies that aK + bK = (a + b)K for all a, b ≥ 0, although this does not hold for general sets. Two sets K and L are homothetic if K = aL+ x for some a > 0 and some point x ∈ R. The n-dimensional (Euclidean) volume of K will be denoted by Vn(K). The Brunn-Minkowski Theorem asserts that the n-th root of the Euclidean volume Vn is concave with respect to Minkowski combinations; that is, for λ ∈ [0, 1], (1) Vn((1− λ)K + λL) ≥ (1− λ)Vn(K) + λVn(L). If K and L have non-empty interiors, then equality holds for some λ ∈ (0, 1) if and only if K and L are homothetic. This article describes a new proof of this equality condition, using a homothetic projection theorem of Hadwiger. The Brunn-Minkowski Theorem is the centerpiece of modern convex geometry [1, 6, 14, 15]. This theorem encodes as special cases the classical isoperimetric inequality (relating volume and surface area [14, p. 318]), Urysohn’s inequality (relating volume and mean width, and strengthening the isodiametric inequality [14, p. 318]), and families of inequalities relating mean projections [14, p. 333]. The concavity implied by (1) leads to families of second-order discriminant-type inequalities for mixed volumes, such as Minkowski’s second mixed volume inequality [14, p. 317] and (after substantial additional labor) the Alexandrov-Fenchel Received by the editors May 9, 2010 and, in revised form, September 2, 2010. 2010 Mathematics Subject Classification. Primary 52A20, 52A38, 52A39, 52A40. c ©2011 American Mathematical Society Reverts to public domain 28 years from publication