17.8 ENTROPY POWER INEQUALITY AND BRUNN–MINKOWSKI INEQUALITY 675
alternative proof of the entropy power inequality. We also show how the
entropy power inequality and the Brunn–Minkowski inequality are related
by means of a common proof.
We can rewrite the entropy power inequality for dimension n = 1in
a form that emphasizes its relationship to the normal distribution. Let
X and Y be two independent random variables with densities, and let
X
and Y
be independent normals with the same entropy as X and
Y , respectively. Then 2
2h(X)
= 2
2h(X
)
= (2πe)σ
2
X
and similarly, 2
2h(Y )
=
(2πe)σ
2
Y
. Hence the entropy power inequality can be rewritten as
2
2h(X+Y)
≥ (2πe)(σ
2
X
+ σ
2
Y
) = 2
2h(X
+Y
)
, (17.89)
since X
and Y
are independent. Thus, we have a new statement of the
entropy power inequality.
Theorem 17.8.1 (Restatement of the entropy power inequality ) For
two independent random variables X and Y ,
h(X + Y) ≥ h(X
+ Y
), (17.90)
where X
and Y
are independent normal random variables with h(X
) =
h(X) and h(Y
) = h(Y ).
This form of the entropy power inequality bears a striking resemblance
to the Brunn–Minkowski inequality, which bounds the volume of set
sums.
Definition The set sum A + B of two sets A, B ⊂
R
n
is defined as the
set {x + y : x ∈ A, y ∈ B}.
Example 17.8.1 The set sum of two spheres of radius 1 is a sphere of
radius 2.
Theorem 17.8.2 (Brunn–Minkowski inequality) The volume of the set
sum of two sets A and B is greater than the volume of the set sum of two
spheres A
and B
with the same volume as A and B, respectively:
V(A+ B) ≥ V(A
+ B
), (17.91)
where A
and B
are spheres with V(A
) = V(A) and V(B
) = V(B).
The similarity between the two theorems was pointed out in [104].
A common proof was found by Dembo [162] and Lieb, starting from a