  We solve a randomized version of the following open question: is there a
strictly convex, bounded curve \gamma in the plane such that the number of
rational points on \gamma, with denominator $n$, approaches infinity with $n$?
Although this natural problem appears to be out of reach using current methods,
we consider a probabilistic analogue using a spatial Poisson-process that
simulates the refined rational lattice $\frac{1}{d} Z^2$, which we call $M_d$,
for each natural number $d$. The main result here is that with probability 1
there exists a strictly convex, bounded curve \gamma such that the number of
spatial Poisson points on \gamma, with intensity $d$, approaches infinity with
$d$. The methods include the notion of a generalized affine length of a convex
curve, defined by Petrov (2007).
