Gruber P. Convex and Discrete Geometry

8 The Brunn–Minkowski Inequality 167
Using the Prékopa–Leindler inequality 8.14 with λ = 1 2 , Maurey [699] gave a
simple proof of the following weaker estimate. A version of Maurey’s argument for
the Euclidean case appeared in an article of Arias de Reyna, Ball and Villa [36], see
also Matouˇsek’s book [695].
Theorem 8.16. Let Ed be endowed with the standard Gaussian measure µ. Then, if
C ⊆ Ed is closed with µ(C) >0 :
�
(3) e 1 4 δC (x) 2
dµ(x) ≤ 1
µ(C) ,
E d
where δC(x) = dist(x, C) = min{�x − y� :y ∈ C}. If, in particular, µ(C) = 1 2 ,
then
ε2
−
(4) µ(Cε) ≥ 1 − 2e 4 .
Proof. Recall the Prékopa–Leindler theorem and let
f (x) = e 1 4 δC (x) 2
δ(x), g(x) = 1C(x)δ(x), h(x) = δ(x) for x ∈ E d .
Here, δ(·) is the density of the Gaussian measure µ, see (2), and 1C the characteristic
function of C. Wehavetoprovethat
�
(5) e 1 4 δC (x) 2
dµ(x) µ(C) ≤ 1,
or
(6)
E d
�
E d
�
�
f (x) dx g(x) dx ≤
�
E d
E d
�2 h(x) dx .
It is thus enough to check that f, g, h, λ= 1 2 satisfy the assumption of the Prékopa–
Leindler inequality, i.e.
�
x + y
�2 (7) f (x) g(y) ≤ h for x, y ∈ E
2
d .
It is sufficient to show (7) for y ∈ C, since otherwise g(y) = 0. But in this case
δC(x) ≤�x − y�. Hence
(2π) d f (x) g(y) = e 1 4 δC (x) 2
e − 1 2 x2
e − 1 2 y2
≤ e 1 4 �x−y�2 − 1 2 �x�2− 1 2 �y�2
= e − 1 4 �x+y�2
= � e − 1 x+y
2 � 2 �2�2 d
= (2π) h
� x + y
This settles (7). By the Prékopa–Leindler theorem, (7) yields (6) which, in turn,
implies (5), concluding the proof of (3).
To obtain the inequality (4) from (3), note that, in case µ(C) = 1 2 , the inequality
(3) implies �
E d
e 1 4 δC (x) 2
dµ(x) ≤ 2.
2
� 2
.