We show that every (possibly unbounded) convex polygon $P$ in $R^2$ with $m$
edges can be represented by inequalities $p_1 \ge 0,...,p_n \ge 0,$ where the
$p_i$'s are products of at most $k$ affine functions each vanishing on an edge
of $P$ and $n=n(m,k)$ satisfies $s(m,k) \le n(m,k) \le (1+\epsilon_m) s(m,k)$
with $s(m,k):=\max \{m/k,\log_2 m\}$ and $\epsilon_m \to 0$ as $m \to \infty$.
This choice of $n$ is asymptotically best possible. An analogous result on
representing the interior of $P$ in the form $p_1 > 0,..., p_n > 0$ is also
given.