Concentration of mass and central limit properties of isotropic convex bodies
Authors: Paouris G. 
Published: October 2004

We discuss the following question: Do there exist an absolute constant

c
>
0

c>0

and a sequence

ϕ
(
n
)

\phi (n)

tending to infinity with

n
n

, such that for every isotropic convex body

K
K

in

R

n

{\mathbb R}^n

and every

t
≥
1

t\geq 1

the inequality
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="Prob left-parenthesis StartSet x element-of upper K colon double-vertical-bar x double-vertical-bar Subscript 2 Baseline greater-than-or-equal-to c StartRoot n EndRoot upper L Subscript upper K Baseline t EndSet right-parenthesis less-than-or-equal-to exp left-parenthesis minus phi left-parenthesis n right-parenthesis t right-parenthesis">

Prob

(

{

x
∈
K
:
‖
x

‖
2

≥
c

n

L
K

t

}

)

≤
exp
⁡

(

−
ϕ
(
n
)
t

)

\textrm {Prob}\left (\big \{ x\in K:\| x\|_2\geq c\sqrt {n}L_Kt\big \}\right ) \leq \exp \big (-\phi (n)t\big )

holds true? Under the additional assumption that

K
K

is 1-unconditional, Bobkov and Nazarov have proved that this is true with

ϕ
(
n
)
≃

n

\phi (n)\simeq \sqrt {n}

. The question is related to the central limit properties of isotropic convex bodies. Consider the spherical average
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="f Subscript upper K Baseline left-parenthesis t right-parenthesis equals integral Underscript upper S Superscript n minus 1 Baseline Endscripts StartAbsoluteValue upper K intersection left-parenthesis theta Superscript up-tack Baseline plus t theta right-parenthesis EndAbsoluteValue sigma left-parenthesis d theta right-parenthesis">

f
K

(
t
)
=

∫

S

n
−
1

|

K
∩
(

θ

⊥

+
t
θ
)

|

σ
(
d
θ
)

f_K(t)=\int _{S^{n-1}}|K\cap (\theta ^{\perp }+t\theta )|\sigma (d\theta )

. We prove that for every

γ
≥
1

\gamma \geq 1

and every isotropic convex body

K
K

in

R

n

{\mathbb R}^n

, the statements (A) “for every

t
≥
1

t\geq 1

,
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="Prob left-parenthesis StartSet x element-of upper K colon double-vertical-bar x double-vertical-bar Subscript 2 Baseline greater-than-or-equal-to gamma StartRoot n EndRoot upper L Subscript upper K Baseline t EndSet right-parenthesis less-than-or-equal-to exp left-parenthesis minus phi left-parenthesis n right-parenthesis t right-parenthesis">

Prob

(

{

x
∈
K
:
‖
x

‖
2

≥
γ

n

L
K

t

}

)

≤
exp
⁡

(

−
ϕ
(
n
)
t

)

\textrm {Prob}\left (\big \{ x\in K:\|x\|_2\geq \gamma \sqrt {n}L_Kt\big \}\right )\leq \exp \big (-\phi (n)t\big )

" and (B) “for every

0
>
t
≤

c
1

(
γ
)

ϕ
(
n
)

L
K

0>t \leq c_1(\gamma )\sqrt {\phi (n)}L_K

,
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="f Subscript upper K Baseline left-parenthesis t right-parenthesis less-than-or-equal-to StartFraction c 2 Over upper L Subscript upper K Baseline EndFraction exp left-parenthesis minus t squared slash left-parenthesis c 3 left-parenthesis gamma right-parenthesis squared upper L Subscript upper K Superscript 2 Baseline right-parenthesis right-parenthesis">

f
K

(
t
)
≤

c
2

L
K

exp
⁡

(

−

t
2

/

(

c
3

(
γ

)
2

L
K
2

)

)

f_K(t)\leq \frac {c_2}{L_K}\exp \big (-t^2/(c_3(\gamma )^2L_K^2)\big )

, where

c
i

(
γ
)
≃
γ

c_i(\gamma )\simeq \gamma

" are equivalent.