The Brunn–Minkowski inequality

Let K and L be compact convex bodies with nonempty interior in the same real d-dimensional Euclidean space. For a number λ between zero and one, their Minkowski interpolation consists of all (1 − λ)x + λy with x in K and y in L. Brunn–Minkowski lower-bounds the volume root of this interpolation by the corresponding weighted average of the two original volume roots. This is a theorem, not a consequence of pointwise convexity alone.

The unweighted version says the volume root of K + L is at least the sum of the individual volume roots. Scaling reduces one version to the other. In dimension one, compact convex bodies are intervals, and their lengths add exactly. In higher dimensions the sum can gain extra volume because the two bodies occupy different directions. The root cannot be omitted from the concavity statement: squares of sides one and two have a midpoint square of area 2.25, below the average input area 2.5, while the corresponding area roots satisfy equality.

For positive-volume convex bodies and an interior weight zero less than λ less than one, equality holds exactly when the two bodies are homothetic: one is a positive dilation and translation of the other. If L = sK + v with s positive, convexity gives (1 − λ)K + λL = ((1 − λ) + λs)K + λv. Taking volume roots then gives equality directly. The necessity direction is a substantive part of the equality theorem, not proved by this calculation.

Equal volumes alone do not force equality. A long horizontal rectangle and a long vertical rectangle may have the same area but create a larger-area average. Translations, in contrast, do not affect volumes or the homothety condition. Degenerate zero-volume sets require a separate equality discussion, so we do not extend the positive-volume characterization to them. At endpoint weights zero or one, equality is automatic regardless of shapes and therefore carries no structural information.

Add a small Euclidean ball εB to K, where B is the unit ball. Steiner’s volume expansion identifies the derivative at ε = 0 with the surface area S(K). Brunn–Minkowski says the volume of the enlarged body is at least the d-th power of |K| to the power 1/d plus ε times the unit-ball volume to the power 1/d. Differentiating this lower bound at zero yields the familiar isoperimetric comparison with a ball of the same volume.

This argument explains why addition of shapes can control boundary size, but it is a consequence sketch, not a complete proof of Brunn–Minkowski. The volume-polynomial facts justify the first-order step for convex bodies, including polytopes with nonsmooth boundaries. For dimension at least two, the result is S(K) at least d times κ_d to the power 1/d times |K| to the power (d − 1)/d. Balls attain equality. The important connection is between how fast a parallel body gains volume and how much boundary it initially has.