  We show that the neutron skin thickness $\Delta r_{np}$ of heavy nuclei is
uniquely fixed by the symmetry energy density slope $L({\rho})$ at a
subsaturation cross density $\rho_c \approx 0.11$ fm$^{-3}$ rather than at
saturation density $\rho_0$, while the binding energy difference $\Delta E$
between a heavy isotope pair is essentially determined by the magnitude of the
symmetry energy $E_{\text{sym}}({\rho})$ at the same $\rho_c$. Furthermore, we
find a value of $L({\rho_c})$ leads to a negative
$E_{\text{sym}}({\rho_{0}})$-$L({\rho_{0}})$ correlation while a value of
$E_{\text{sym}}({\rho_{c})}$ leads to a positive one. Using data on $\Delta
r_{np}$ of Sn isotopes and $\Delta E$ of a number of heavy isotope pairs, we
obtain simultaneously $E_{\text{sym}}({\rho_{c})}=26.65\pm0.20$ MeV and
$L({\rho_c})= 46.0\pm4.5$ MeV at 95% confidence level, whose extrapolation
gives $E_{\text{sym}}({\rho_{0}})=32.3\pm1.0$ MeV and
$L({\rho_{0}})=45.2\pm10.0$ MeV. The implication of these new constraints on
the $\Delta r_{np}$ of $^{208}$Pb and the core-crust transition density in
neutron stars is discussed.
