Lemma 6.7.4(Symmetrization with Gaussians). Let $X_1,\dots,X_N$ be independent, mean zero random vectors in a normed space. Let $g_1,\dots,g_N \sim N(0,1)$ be independent Gaussian random variables, which are also independent of $X_i$. Then
$$
\frac{c}{\sqrt{logN}} \mathbb{E} \bigg\|\sum_{i=1}^N g_iX_i \bigg\| \leq \mathbb{E} \bigg \| \sum_{i=1}^N X_i \bigg\| \leq 3 \mathbb{E} \bigg \| \sum_{i=1}^N g_iX_i \bigg \|.
$$
Exercise 6.7.5
Show that the factor $\sqrt{logN}$ in Lemma 6.7.4 is neeed in general, and is optimal. Thus, symmetrization with Gaussian random variables is generally weaker than symmetrization with symmetric Bernoullis.
I cannot find the way to solve this exercise.
Best Answer
Take the normed space to be ${\mathbb R}^N$ with the $\ell_\infty$ norm, and denote by $\{e_j\}_{j=1}^N$ the standard basis of ${\mathbb R}^N$. Let each variable $X_i$ take one of the two values $\pm e_i$ with equal probability. Then the result follows because the maximum of $N$ independent standard Gaussians is asymptotic to $\sqrt{2\log N}$.
See, e.g. the solved exercise 18.7 (pages 332 and 358) in https://homes.cs.washington.edu/~karlin/GameTheoryBook.pdf
** Refinement** To get an example where the $X_i$ are i.i.d., take the normed space to be ${\mathbb R}^{N^3}$ with the $\ell_\infty$ norm, and let each variable $X_i$ be uniformly distributed among the $2N^3$ vectors $\{\pm e_j : j=1,\ldots,N^3\}$.