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 chai
ns 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.
