Concentration of mass and central limit properties of isotropic convex bodies

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 ${\rm Prob}\left (\big \{ x\in K:\Vert x\Vert _2\geq c\sqrt{n}L_Kt\big \}\right ) \leq\exp \big (-\phi (n)t\big )$ 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 $f_K(t)=\int_{S^{n-1}}\vert K\cap (\theta^{\perp }+t\theta )\vert\sigma (d\theta )$. We prove that for every $\gamma\geq 1$ and every isotropic convex body $K$in ${\mathbb R}^n$, the statements (A) ``for every $t\geq 1$, ${\rm Prob}\left (\big\{ x\in K:\Vert x\Vert _2\geq \gamma\sqrt{n}L_Kt\big\}\right )\leq\exp \big (-\phi (n)t\big )$" 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.