Summary:
ABSTRACT
The main theme of this Ph.D. Thesis is the use of geometric and probabilistic
methods
for the study of the geometry of logarithmically concave measures in high
dimensions.
We discuss the following topics of the theory:
1. -estimates for random marginals. Let (
;A; ) be a probability space. For
any function f : (
; ) ! R which is A-measurable, we de ne the -norm of f (1 6
6 2) as follows:
kfk = inf
t > 0 :
Z
exp
jf(!)j
t
d(!) 6 2
:
Let be a log-concave Borel probability measure on Rn and let 2 [1; 2]. We
say
that satis es a estimate in the direction of 2 Sn1 if there exists a
constant
b = b () > 0 such that kh; ik 6 b kh; ik2. We say that is a
measure with
constant B if B := sup2Sn1 b () < 1.
For any subspace F of Rn we de ne the projection (marginal) F () of with
F ()(A) := (P1
F (A)) for any Borel set A in F. It is known that every log-concave
probability measure is a 1-measure with constant B1() 6 C, where C > 0 is an
absolute constant. We show that a random marginal F () of an isotropic
log-concave
probability measure on Rn exhibits better -behavior. For a natural variant
0
of
the standard -norm we show the following:
(i) If k 6 p
n, then for a random F 2 Gn;k we have that F () is a 02
-measure.
We complement this result by showing that a random F () is, at the same time,
supergaussian.
(ii) If k = n, 1
2 < < 1, then for a random F 2 Gn;k we have that F () is a
0
()-measure, where () = 2
31 .
2. Subgaussian directions of log-concave measures. Let be a log-concave
probability measure on Rn. A direction 2 Sn1 is called subgaussian (with
constant
b > 0) for if the following estimate holds:
kh; ik 2 6 bkh; ik2:
We show that if is a centered log-concave probability measure on Rn then,
c1 p
n 6 j 2()j1=n 6 c2
p
log n
p
n
;
where 2() is the 2-body of de ned by its support function h 2()()
:= kh; ik 2 ; 2
Sn1 and c1; c2 > 0 are absolute constants. A direct consequence of the
previous volumetric
estimate is the existence of subgaussian directions for with constant r =
O(
p
log n).
Using the basic argument of the proof \hereditarily", we can gain some extra
information
on the distribution of the 2-norm of linear functionals on isotropic convex
bodies. In particular, we can show the following measure estimate: If K is an
isotropic
convex body on Rn then
(f 2 Sn1 : kh; ik 2 6 ct
p
log nLKg) > ecn=t2
;
for all t > 1, where c > 0 is an absolute constant. For larger values of t a
better estimate
is provided. As an application we provide a dichotomy result for the problem of
giving
an upper bound for the mean width of an isotropic convex body: For any 2 6 q 6
n we
de ne the Dvoretzky numbers of the Lq-centroid bodies of K:
k(q) := n
w(Zq(K))
R(Zq(K))
2
:
We set = (K) := min26q6n k(q) and we prove that
w(K) 6 C
p
n minf
p
;
p
n=gLK;
where C > 0 is an absolute constant. From the above estimate we recover the,
presently,
best (general) upper bound for the mean width of an isotropic convex body.
3. Log-concave measures satisfying logarithmic Sobolev inequality. Let be
a Borel probability measure on Rn. We say that satis es the log-Sobolev
inequality
with constant > 0 if for any (locally) Lipschitz function f : Rn ! R we have
Ent(f2) 6 2
Z
Rn
krfk22
d;
where Ent(g) is the entropy of g with respect to : for any g : Rn ! R+ we de
ne
Ent(g) := E(g log g) E(g) log E(g):
Starting with the observation that a log-concave isotropic measure on Rn
which satis es
the log-Sobolev inequality with constant is 2 with constant b = O(
p
), we prove that
it shares many of the geometric properties discussed in the previous Chapters.
Finally,
we show that a log-concave measure which satis es the log-Sobolev inequality
with
constant , also has property ( ) with cost function w(x) = c
kxk22
, i.e. for any bounded
measurable function f on Rn one has
Z
Rn
ef2w d
Z
Rn
ef d 6 1;
where f2w is the minimal convolution of f and w de ned by
(f2w)(x) := infff(y) + w(x y) : y 2 Rng:
Keywords:
Logarithmically concave measure, Isotropic convex bodies, \psi_\alpha - estimates, Log-Sobolev inequality, Infimum convolution
