  We study the convex hull of the first $n$ steps of a planar random walk, and
present large-$n$ asymptotic results on its perimeter length $L_n$, diameter
$D_n$, and shape. In the case where the walk has a non-zero mean drift, we show
that $L_n / D_n \to 2$ a.s., and give distributional limit theorems and
variance asymptotics for $D_n$, and in the zero-drift case we show that the
convex hull is infinitely often arbitrarily well-approximated in shape by any
unit-diameter compact convex set containing the origin, and then $\liminf_{n
\to \infty} L_n/D_n =2$ and $\limsup_{n \to \infty} L_n /D_n = \pi$, a.s. Among
the tools that we use is a zero-one law for convex hulls of random walks.
