The Brunn-Minkowski Inequality
Oded Stein, Wu Jiqing and Giuliano Basso
December 18, 2014
The Isoperimetric Inequality
Definition 1. The surface area of a convex compact nonempty K ⊆ Rn is
defined as follows:
Sn (K) := lim
ε→0+
Vn (K + εB) − Vn (K)
ε
where B is the n-dimensional unit ball.
Theorem 1. (Isoperimetric Inequality)1 For a convex compact nonempty
K ⊆ Rn it holds:
1
1
S(K) n−1
V(K) n
6
V(B)
S(B)
1
The isoperimetric inequality follows
from the Brunn-Minkowski inequality
where B is the n-dimensional unit ball.
The Brunn-Minkowski theorem
For all K0 , K1 ∈ Kn , s, t > 0 and λ ∈ [0, 1] we have the following
equivalent statements 2 :
Vn ((1 − λ)K0 + λK1 )1/n > (1 − λ)Vn (K0 )1/n + λVn (K1 )1/n
2
(1)
Vn (K0 + K1 )1/n > Vn (K0 )1/n + Vn (K1 )1/n
(2)
Vn (sK0 + tK1 )1/n > sVn (K0 )1/n + tVn (K1 )1/n
(3)
Vn (K0 ) = Vn (K1 ) = 1 ⇒ Vn ((1 − λ)K0 + λK1 ) > 1
(4)
Vn ((1 − λ)K0 + λK1 ) > min{Vn (K0 ), Vn (K1 )}
(5)
Vn ((1 − λ)K0 + λK1 ) > Vn (K0 )1−λ Vn (K1 )λ
(6)
Theorem 2. (Prékopa-Leindler)3 Let λ ∈ (0, 1), and f, g, h be nonnegative,
Lebesgue integrable real functions on Rn with
h((1 − λ)x + λy) > f(x)1−λ g(y)λ .
The proof of the equivalent relations
is straightforward, as to Ineq. (6) the
arithmetic-geometric mean inequality
will do the job.
3
The generalized Brunn-Minkowski
inequality is the immediate conclusion
of Prékopa-Leindler inequality. How?
the brunn-minkowski inequality
2
Then
Z
Rn
Z
h>
Rn
1−λ Z
f
Rn
λ
g
.
Theorem 3. (Brunn-Minkowski)4 For K0 , K1 ∈ Kn and for λ ∈ [0, 1]
Ineq. (1) holds, equality for λ ∈ (0, 1) holds iff K0 , K1 either lie in parallel
hyperplanes or are homothetic.
Applications and extensions
4
Sketch of proof:
Case 1:K0 , K1 lie in parallel hyperplanes or homothetic.
Case 2: dim(K0 ), dim(K1 ) < n.
Case 3: dim(K0 ) < n, dim(K1 ) = n.
Case 4: dim(K0 ), dim(K1 ) = n. By
reducing the problem to Ineq. (4) and
induction we prove the theorem.
Let K, L ∈ Kn denote two convex bodies. The existence of the following limit
nV(K, n − 1; L) := lim
ε→0+
Vn (K + εL) − Vn (K)
ε
follows from Minkowski’s theorem on mixed volumes. 5
Theorem 4. (Mikowski’s first inequality) Let K, L ∈ Kn be two convex
bodies. Then
n−1
1
V(K, n − 1; L) > Vn (K) n Vn (L) n ,
5
On account of Minkowski’s theorem
on mixed
we have Vn (K +
P volumes
n
n−i .
εL) = n
i=0 i V(K, i; L, n − i)ε
Now rearrange.
with equality if and only if K, L are homothetic.
The Minkowski sum of two Lebesgue measurable subsets of Rn is
in general not Lebesgue measurable. Sierpiński found a counterexample for the case n = 1, cf. 6 .
Definition 2. (Essential sum) Let A, B ⊂ Rn denote two subsets of Rn .
We call the set
6
Waclaw Sierpiński. Sur la question de
la mesurabilité de la base de M. Hamel.
Fundamenta Mathematicae, 1(1):105–111,
1920
ess(A + B) := {x ∈ Rn | Vn ((x − A) ∩ B) > 0}
the essential sum of A and B.
The following Theorem is due to Brascamp and Lieb, cf. 7 .
Theorem 5. (Brascamp-Lieb) Let A, B ⊂ Rn be two Lebesgue measurable
subsets with Vn (A), Vn (B) > 0. Then ess(A + B) is open and
1
1
1
Vn (ess(A + B)) n > Vn (A) n + Vn (B) n .
References
Herm Brascamp and Elliott Lieb. On extensions of the BrunnMinkowski and Prékopa-Leindler theorems, including inequalities
for log concave functions, and with an application to the diffusion
equation. Journal of Functional Analysis, 22(4):366 – 389, 1976.
Waclaw Sierpiński. Sur la question de la mesurabilité de la base de
M. Hamel. Fundamenta Mathematicae, 1(1):105–111, 1920.
7
Herm Brascamp and Elliott Lieb. On
extensions of the Brunn-Minkowski and
Prékopa-Leindler theorems, including
inequalities for log concave functions,
and with an application to the diffusion
equation. Journal of Functional Analysis,
22(4):366 – 389, 1976