  Let ${\varPi}_n$ be the set of convex polygonal lines $\varGamma$ with
vertices on $\mathbb {Z}_+^2$ and fixed endpoints $0=(0,0)$ and $n=(n_1,n_2)$.
We are concerned with the limit shape, as $n\to\infty$, of "typical"
$\varGamma\in {\varPi}_n$ with respect to a parametric family of probability
measures $\{P_n^r,0<r<\infty\}$ on ${\varPi}_n$, including the uniform
distribution ($r=1$) for which the limit shape was found in the early 1990s
independently by A. M. Vershik, I. B\'ar\'any and Ya. G. Sinai. We show that,
in fact, the limit shape is universal in the class $\{P^r_n\}$, even though
$P^r_n$ ($r\ne1$) and $P^1_n$ are asymptotically singular. Measures $P^r_n$ are
constructed, following Sinai's approach, as conditional distributions
$Q_z^r(\cdot |{\varPi}_n)$, where $Q_z^r$ are suitable product measures on the
space ${\varPi}=\bigcup_n{\varPi}_n$, depending on an auxiliary "free"
parameter $z=(z_1,z_2)$. The transition from $({\varPi},Q_z^r)$ to
$({\varPi}_n,P_n^r)$ is based on the asymptotics of the probability
$Q_z^r({\varPi}_n)$, furnished by a certain two-dimensional local limit
theorem. The proofs involve subtle analytical tools including the M\"obius
inversion formula and properties of zeroes of the Riemann zeta function.
