The concentration of measure on the cube $ [0, 1]^n $ equipped with uniform probability measure $\mu_{\infty}$, states that for any $A \subset [0, 1]^n $ with $ \mu_{\infty}(A) \geq \frac{1}{2} $, we have: \begin{align} 1 - \mu_{\infty}(A_{\epsilon}) \leq e^{- \pi \epsilon^2 }, \epsilon > 0 \end{align}

where $A_{\epsilon} = \{ x \in [0, 1]^n; d(x, A) < \epsilon \}$, and $d = L_2$ is the Euclidean distance. Similarly, let $B^n_{2}$ denote the Euclidean ball with normalized uniform measure. Then there is a constant $C$, for any $A \subset B^n_{2} $, with $ \mu_2(A) > 0 $:

$$ 1 - \mu_2(A_{\epsilon}) \leq \frac{1}{\mu_2(A)} e^{-n C\epsilon^{2} } $$ where $A_{\epsilon} = \{ x \in B^n_2; d(x, A) < \epsilon \}$, and $d = L_2$ is again the $L^2$ norm.

The concentration rate on $ [0, 1]^n $ is slower by a factor of $n$ compared to $B^n_2$. I'm wondering if there is a family of convex bodies $\Gamma(t)$, $ t \in [0, 1] $ with the following properties:

- $\Gamma(0) = [0, 1]^n $ and $\Gamma(1) = B^n_2 $,
- $ \Gamma(t) $ are all equipped with normalized uniform measures, and Euclidean $L_2$ metric,
- each $ \Gamma(t) $ has a concentration rate $\gamma_t(n, \epsilon)$ such that $\gamma_0(n, \epsilon) = \pi \epsilon^2 $ (for the cube) and $\gamma_1(n, \epsilon) = n C \epsilon^2 $ (for the $L^2$ ball),
- As $t$ goes from $0$ to $1$, $\gamma_t(n)$ "concentrates faster". So we get a family of concentration inequalities, interpolating between the ones on $[0, 1]^n $ and $B^n_2$.

A natural example is the concentration of measures on $L^p$ balls, but the version I know is equiped with $d = L_p$ metric, instead of $L_2$. Roughly, I want to know if it is true that the more symmetric a convex body is, the faster its concentration rate.