Let A,B⊆**R**^{d} be two *compact* bodies. Then

(vol(A+B))^{1/d} ≥ (vol(A))^{1/d} + (vol(B))^{1/d},

where vol is

*volume* in

**R**^{d} 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 **R**^{d} by t, its volume grows by t^{d}; the d'th root makes vol(tA)^{1/d} grow linearly.

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

(vol(C_{t}))^{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⊆**R**^{d+1}. Draw an axis through K, and consider the d-dimensional "slice" K_{x} 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(K_{t}) (i.e. it first increases, then decreases). Indeed, for some 0<t<1 y=tx+(1-t)z. Convexity of K implies that t K_{x} + (1-t) K_{z} ⊆ K_{y}. So

(vol(K_{y}))^{1/d} ≥
(vol(t K_{x} + (1-t) K_{z}))^{1/d} ≥

t (vol(K_{x}))^{1/d} + (1-t) (vol(K_{z}))^{1/d} ≥
(min(vol(K_{x}),vol(K_{z}))^{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.