I am self-studying Superconcentration and Related Topics by Sourav Chatterjee for my research and I would like to use the Sudakov minoration in Lemma A.3 in my work. It is stated below:
Let $g$ be a centered Gaussian n-vector. Suppose $\alpha$ is a constant such that $\mathbb{E}(g_i−g_j)^2 \geq \alpha$ for all $i\neq j$. Then $\mathbb{E}[\max_{i} g_i] \geq C\alpha \sqrt{\log n}$, where C is a positive universal constant.
I cannot find references or estimates for actual values of C anywhere. In The Generic Chaining:Upper and Lower Bounds of Stochastic Processes by Michel Talagrand, he states in Lemma 2.1.2 that the value of the constant C above "...remain[s] the same (at least within the same section)".
What does this statement mean? If the constant is universal/absolute, then does its value stay the same no matter the Gaussian process considered? Where do I find values for the constant C?