Let A,BRd be two compact bodies. Then

(vol(A+B))1/d ≥ (vol(A))1/d + (vol(B))1/d,
where vol is volume in Rd and A+B={a+b:a∈A,b∈B} is the (Minkowski) sum of the 2 bodies.

The d'th root may seem a bit surprising. But dimensional analysis shows it makes sense: if we multiply a body in Rd by t, its volume grows by td; the d'th root makes vol(tA)1/d grow linearly.

If we take a "convex combination" Ct=tA+(1-t)B of A and B, we find that

(vol(Ct))1/d = (vol(tA+(1-t)B))1/d ≥ t (vol(A))1/d + (1-t) (vol(B))1/d
which looks nice.

Example (and application)

Suppose now that we have some convex body K⊆Rd+1. Draw an axis through K, and consider the d-dimensional "slice" Kx of K perpendicular to the axis at point x. If x,y,z occur in that order along the axis, then we claim that we have a unimodularity condition on the function vol(Kt) (i.e. it first increases, then decreases). Indeed, for some 0<t<1 y=tx+(1-t)z. Convexity of K implies that t Kx + (1-t) Kz ⊆ Ky. So

(vol(Ky))1/d ≥ (vol(t Kx + (1-t) Kz))1/d
t (vol(Kx))1/d + (1-t) (vol(Kz))1/d ≥ (min(vol(Kx),vol(Kz))1/d.
Raising both sides to the d'th power, we see that vol(Ky) is at least as large as one of vol(Kx),vol(Kz) -- unimodularity.

This is in fact a proof of Brunn's inequality.

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