Generalized Sudakov and dual Sudakov minoration conjecture

Let $u$ be an origin-symmetric \log-concave probability measure on $\mathbb{R}^n$ and let $K\subset \mathbb{R}^n$ be an origin-symmetric convex body. For $p\geq 1$, write $B_p(\mu)$ for the associated centroid body, $Z_p(\mu)$ for the $L_p$ centroid body, and define

$$I^*_1(\mu,K):=\int \\|x\\|_K^*\,d\mu(x),\qquad I_1(\mu,K):=\int \\|x\\|_K\,d\mu(x).$$

Generalized Sudakov minoration conjecture. There is a universal constant $C>0$ such that

$$M(K, C I^*_1(\mu,K) B_p(\mu))\leq \exp(Cp),\qquad \forall p\geq 1,$$

and

$$M(Z_p(\mu), C I_1(\mu,K)K)\leq \exp(Cp),\qquad \forall p\geq 1.$$

These conjectural estimates extend the classical Sudakov and dual Sudakov minoration bounds from Gaussian measures to general origin-symmetric log-concave measures. The second inequality is the dual generalized Sudakov conjecture, asserting that $Z_p(\mu)$ can be covered by exponentially many copies of $C I_1(\mu,K)K$ rather than the single, generally larger scale $I_p(\mu,K)K$ supplied by the elementary inclusion.