  Assume $X_n$ is a random sample of $n$ uniform, independent points from a
triangle $T$. The longest convex chain, $Y$, of $X_n$ is defined naturally. The
length $|Y|$ of $Y$ is a random variable, denoted by $L_n$. In this article, we
determine the order of magnitude of the expectation of $L_n$. We show further
that $L_n$ is highly concentrated around its mean, and that the longest convex
chains have a limit shape.
