The relation between Coulomb displacement energies,ΔEc, andΔr=rn-rp, the difference between the rms radii of neutrons and protons in nuclei, is investigated within the energy density formalism (EDF). The variational equation, obtained by minimizing the Coulomb plus symmetry energies, is solved assuming the symmetry interaction is a simple functional of the local nuclear matter density. Varying parameters of the model, rather unique relation betweenΔEc andΔr is obtained (within ±50 keV).ΔEc isindependent ofrex, the rms radius of the excess neutrons distribution. Using the experimental values ofrp and adjusting the model to reproduce the recent data onΔr (Δr∽~0.05 fm for48Ca and208Pb), which are significantly smaller than those obtained from current Hartree-Fock calculations, the calculatedΔEc agree with the experimental results. Using the value ofΔr∼0.05 fm and the experimental values ofrex, a small compression (<0.02 fm) of the proton core in the analogue state relative to its parent state emerges, thus contributing an additional electrostatic term to the Coulomb displacement energy. The size of this relative core-compression effect depends on the values assumed forΔr andrex, it increases with the decreasing ofΔr and the increasing ofrex. IfΔr∼0.05 fm the effect is large enough to remove the long standing Coulomb energy anomaly. The main result of the present work is the correlation betweenΔEc andΔr, suggesting that the difficulties of current Hartree-Fock calculations in reproducing isotope shifts ofrp, the small value ofrn−rp and the values ofΔEc may all be different manifestations of some missing residualp n effective interaction.