It's well-known that the Euclidean $2$-norm of subgaussian random vectors concentrates in high dimensions, e.g. when $X \sim \mathcal{N}(0,I_n),$ (or in general $X$ is subgaussian with independent co-ordinates), $||X||_2$ concentrates near $\sqrt{n}.$ (See Theorem 3.1.1. of this book).
My question is about possible results/papers on concentration on $||X||_p, p \ne 2.$ when $X$ is subgaussian or satisfy some kind of tail decay property like $P[||X||_2 >t]\le e^{-\beta(t)}, \beta(t) > 0,$ for example. More particularly, can we say that the larger the $p,$ the stronger/tighter is the concentration of $||X||_p$ ariund $\mathbb{E}||X||_p?$ So to be precise: can we say $P[| ||X||_p - \mathbb{E}||X||_p | > t] \le P[| ||X||_q - \mathbb{E}||X||_q |> t]$ when $p \ge q$? You can assume first that $X \sim \mathcal{N}(0,I_n),$ or at least $X$ has independent co-ordinates which are sub Gaussian.
The reason is I ask this question is motivated by my study of this paper on machine learning, where they show that for high dimensions, the $L^p$ distance has more discriminative power between nearest and farthest points from the origin, if $p$ is less, and this discriminative power decreases when $p$ increases. See Lemma 1, Corollary 2. They in fact advocate the use of $L^p$ pseudo-metrics for $0<p<1$, for in this case, the discriminative power is high. This led me to guess that for Gaussian or subgaussian random vectors, the concentration of $p$-norm is more tight if $p$ is large, and vice versa. Thanks in advance!
