  Self-avoiding polymers in two-dimensional ($d=2$) melts are known to adopt
compact configurations of typical size $R(N) \sim N^{1/d}$ with $N$ being the
chain length. Using molecular dynamics simulations we show that the irregular
shapes of these chains are characterized by a perimeter length $L(N) \sim
R(N)^{\dpm}$ of fractal dimension $\dpm = d-\Theta_2 =5/4$ with $\Theta_2=3/4$
being a well-known contact exponent. Due to the self-similar structure of the
chains, compactness and perimeter fractality repeat for subchains of all
arc-lengths $s$ down to a few monomers. The Kratky representation of the
intramolecular form factor $F(q)$ reveals a strong non-monotonous behavior with
$q^2F(q) \sim 1/(qN^{1/d})^{\Theta_2}$ in the intermediate regime of the
wavevector $q$. Measuring the scattering of labeled subchains %($s F(q) \sim
L(s)$) the form factor may allow to test our predictions in real experiments.
