Let A,B⊆Rd 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.
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(K
y) is at least as large as one of vol(K
x),vol(K
z) --
unimodularity.
This is in fact a proof of Brunn's inequality.