  Consider a dataset of n(d) points generated independently from R^d according
to a common p.d.f. f_d with support(f_d) = [0,1]^d and sup{f_d([0,1]^d)}
growing sub-exponentially in d. We prove that: (i) if n(d) grows
sub-exponentially in d, then, for any query point q^d in [0,1]^d and any
epsilon>0, the ratio of the distance between any two dataset points and q^d is
less that 1+epsilon with probability -->1 as d-->infinity; (ii) if
n(d)>[4(1+epsilon)]^d for large d, then for all q^d in [0,1]^d (except a small
subset) and any epsilon>0, the distance ratio is less than 1+epsilon with
limiting probability strictly bounded away from one. Moreover, we provide
preliminary results along the lines of (i) when f_d=N(mu_d,Sigma_d).
