Concentration of mass and central limit properties of isotropic convex bodies

Paouris, G.

doi:10.1090/S0002-9939-04-07757-3

Concentration of mass and central limit properties of isotropic convex bodies
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Proc. Amer. Math. Soc. 133 (2005), 565-575 Request permission

Abstract:

We discuss the following question: Do there exist an absolute constant $c>0$ and a sequence $\phi (n)$ tending to infinity with $n$, such that for every isotropic convex body $K$ in ${\mathbb R}^n$ and every $t\geq 1$ the inequality $\textrm {Prob}\left (\big \{ x\in K:\| x\|_2\geq c\sqrt {n}L_Kt\big \}\right ) \leq \exp \big (-\phi (n)t\big )$ holds true? Under the additional assumption that $K$ is 1-unconditional, Bobkov and Nazarov have proved that this is true with $\phi (n)\simeq \sqrt {n}$. The question is related to the central limit properties of isotropic convex bodies. Consider the spherical average $f_K(t)=\int _{S^{n-1}}|K\cap (\theta ^{\perp }+t\theta )|\sigma (d\theta )$. We prove that for every $\gamma \geq 1$ and every isotropic convex body $K$ in ${\mathbb R}^n$, the statements (A) “for every $t\geq 1$, $\textrm {Prob}\left (\big \{ x\in K:\|x\|_2\geq \gamma \sqrt {n}L_Kt\big \}\right )\leq \exp \big (-\phi (n)t\big )$" and (B) “for every $0<t \leq c_1(\gamma )\sqrt {\phi (n)}L_K$, $f_K(t)\leq \frac {c_2}{L_K}\exp \big (-t^2/(c_3(\gamma )^2L_K^2)\big )$, where $c_i(\gamma )\simeq \gamma$" are equivalent.

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