Gruber P. Convex and Discrete Geometry

166 Convex Bodies
Since f has Lipschitz constant 1, f (u) ≤ m + ε for u ∈ Cε and f (v) ≥ m − ε for
v ∈ Dε. This shows that for large d
S � {u ∈ S d−1 :|f (u) − m| >ε} � is at most about 2e − 1 2 dε2
.
In other words, for large d a real function on S d−1 with Lipschitz constant 1 is nearly
equal to its median on most of S d−1 .
Heuristic Observations
This surprising phenomenon appears in one form or another in many results of the
local theory of normed spaces, as initiated by Milman and his collaborators. Milman
[726] expressed this in his unique way as follows:
This phenomenon led to a complete reversal of our intuition on high-dimensional
results. Instead of a chaotic diversity with an increase in dimension, which previous
intuition suggested, we observe well organized and simple patterns of behaviour.
For similar situations dealing with approximation of convex bodies, the Minkowski–
Hlawka theorem and Siegel’s mean value formula, see Sects. 11.2 and 24.2.
The Case of Gaussian Measure
Let Ed be endowed with the standard Gaussian probability measure µ. It has density:
(2) δ(x) = 1
e − 1 2 �x�2
for x ∈ Ed .
(2π) d 2
Borell [150] showed that, in this space, we have the following Brunn–Minkowski
type result. Let α>0. Then for all closed sets C ⊆ E d with µ(C) = α,
µ(Cε) ≥ µ(H + ε
) for ε ≥ 0,
where H + is a closed halfspace in E d with µ(H + ) = α.
In particular, if α = 1 2 , then for H + we may take the halfspace {x : x1 ≤ 0}.
Then
Hence
µ(E d \H + ε
) = 1
(2π) d 2
= 1
√ 2π
≤ e −ε2
2
�
x1≥ε
�
+∞
0
1
√ 2π
e − x2 1 2 −···− x2 d 2 dx = 1
√ 2π
e −(t+ε)2
2 dt = e −ε2
2
�
+∞
0
1
√ 2π
�
�
+∞
ε
+∞
t2
ε2
−
e 2 −
dt = e 2 for ε ≥ 0.
ε2
−
µ(Cε) ≥ 1 − e 2 for ε ≥ 0.
0
e − x2 1 2 dx1
t2
−
e 2 −tε dt