Let $\{x_1, \dots x_n\} \in \mathbb{R}^{p\times 1}$ be $n$ distinct columns vectors. Consider the $n \times n$ "kernel" matrix used often in machine learning community:
$$K:= [k(x_i,x_j)]_{n \times n}$$
where:
$$k(x_i,x_j):= f(\frac{||x_i-x_j||^2}{p})$$
where $f: [0, \infty)\to [0, \infty)$ be a sufficiently smooth function.
(1) What can we say about $rank(K)$ when $f:= Id$?
(2) What can we say about $rank(K)$ when $f:= 1$ on [0, 1) and $0$ otherwise?
(3) What can we say about $rank(K)$ when $f(r):= e^{-r/2}$? (This is the case of "Gaussian kernel").