The Sudakov inequalities bound the number of scaled Euclidean balls needed to cover a centrally symmetric convex body $K \subset \mathbb{R}^n$, and vice versa. Before I state them, let me say what I'm interested in: the best possible constant $c$ in the exponent, particularly in the primal inequality. (I'm a computer scientist trying to use these to analyze an algorithm.) Thanks in advance!
Here are the Sudakov inequalities, stated (as is typical) without explicit constants ["Asymptotic Geometric Analysis" by Artstein-Avidan, Giannopoulos, and Milman, Theorems 4.2.1 and 4.2.2, https://bookstore.ams.org/surv-202].
Primal Sudakov (Theorem 4.2.1): Let $K$ be a centrally symmetric body in $\mathbb{R}^n$. For every $t > 0$, $$\log N(K, tB^n_2) \leq c n \frac{M(K^\circ)^2}{t^2}.$$
Dual Sudakov (Theorem 4.2.2): Let $K$ be a centrally symmetric body in $\mathbb{R}^n$. For every $t > 0$, $$\log N(B^n_2, tK) \leq cn \frac{M(K)^2}{t^2}.$$
Here, $B_2^n \subset \mathbb{R}^n$ is the Euclidean unit ball, and for sets $U, V \subset \mathbb{R}^n$, $N(U, V)$ is the number of translates of $V$ needed to cover $U$. The letter $c$ stands for an unspecified universal constant. Essentially following the proofs in the reference, I've achieved roughly $c = 8$ in the dual and $c = 800$ in the primal inequality--it'd be really great to improve the latter.
The parameters $M(K)$ and $M(K^\circ)$ are defined as follows. $\| \cdot \|_K$ is the norm associated with $K$, i.e., $$\|x\|_K := \sup \{s \geq 0: x \in s \cdot K\}.$$ $M(K)$ is the expectation of $\|X\|_K$, where $X$ is uniformly distributed on the unit sphere $S_2^{n-1} \subset \mathbb{R}^{n}$. (Note that for centrally symmetric $K$ we have $M(K) = w(K^\circ)/2$, where $w(\cdot)$ is the mean width appearing in the original statements.)
Finally, $K^\circ$ is the polar $$K^\circ := \{x \in \mathbb{R}^n : \forall y \in K, x \cdot y \leq 1\}.$$