  For two convex discs $K$ and $L$, we say that $K$ is $L$-convex if it is
equal to the intersection of all translates of $L$ that contain $K$. In
$L$-convexity the set $L$ plays a similar role as closed half-spaces do in the
classical notion of convexity. We study the following probability model: Let
$K$ and $L$ be $C^2_+$ smooth convex discs such that $K$ is $L$-convex. Select
$n$ i.i.d. uniform random points $x_1,\ldots, x_n$ from $K$, and consider the
intersection $K_{(n)}$ of all translates of $L$ that contain all of
$x_1,\ldots, x_n$. The set $K_{(n)}$ is a random $L$-convex polygon in $K$. We
study the expectation of the number of vertices $f_0(K_{(n)})$ and the missed
area $A(K\setminus K_{n})$ as $n$ tends to infinity. We consider two special
cases of the model. In the first case we assume that the maximum of the
curvature of the boundary of $L$ is strictly less than $1$ and the minimum of
the curvature of $K$ is larger than $1$. In this setting the expected number of
vertices and missed area behave in a similar way as in the classical convex
case and in the $r$-spindle convex case (when $L$ is a radius $r$ circular
disc). The other case we study is when $K=L$. This setting is special in the
sense that an interesting phenomenon occurs: the expected number of vertices
tends to a finite limit depending only on $L$. This was previously observed in
the special case when $L$ is a circle of radius $r$ (Fodor, Kevei and V\'igh
(2014)). We also determine the extrema of the limit of the expectation of the
number of vertices of $L_{(n)}$ if $L$ is a convex discs of constant width $1$.
The formulas we prove can be considered as generalizations of the corresponding
$r$-spindle convex statements proved by Fodor, Kevei and V\'igh (2014).
