  Convex records have an appealing purely geometric definition. In a sequence
of $d$-dimensional data points, the $n$-th point is a convex record if it lies
outside the convex hull of all preceding points. We specifically focus on the
bivariate (i.e., two-dimensional) setting. For iid (independent and identically
distributed) points, we establish an identity relating the mean number
$\mean{R_n}$ of convex records up to time $n$ to the mean number $\mean{N_n}$
of vertices in the convex hull of the first $n$ points. By combining this
identity with extensive numerical simulations, we provide a comprehensive
overview of the statistics of convex records for various examples of iid data
points in the plane: uniform points in the square and in the disk, Gaussian
points and points with an isotropic power-law distribution. In all these cases,
the mean values and variances of $N_n$ and $R_n$ grow proportionally to each
other, resulting in finite limit Fano factors $F_N$ and $F_R$. We also consider
planar random walks, i.e., sequences of points with iid increments. For both
the Pearson walk in the continuum and the P\'olya walk on a lattice, we
characterise the growth of the mean number $\mean{R_n}$ of convex records and
demonstrate that the ratio $R_n/\mean{R_n}$ keeps fluctuating with a universal
limit distribution.
