On the invariant distribution of galaxies in the re-⟨μ⟩e plane out to z=0.64

We study the evolution of the relation between half-light (effective) radius, r(e), and mean surface brightness, [mu](e), known as the Kormendy relation, out to redshift z = 0.64 in the V-band rest frame on the basis of a large sample of spheroidal galaxies (N = 228) belonging to three clusters of galaxies. The present sample constitutes the largest data set for which the Kormendy relation is investigated up to a look-back time of similar to 6 Gyr (H-0 = 70 km s(-1) Mpc(-1), Omega(m) = 0.3, Omega(Lambda) = 0.7). A new fitting procedure, which suitably accounts for selection criteria effects, makes it possible for the first time to study the trend of the slope (beta) and of the intrinsic dispersion (sigma([mu]e)((i))) of the Kormendy relation, and the properties of the whole distribution in the r(e)-[mu](e) plane as a function of look-back time. The slope beta of the relation does not change from z = 0.64 to the present epoch: beta = 2.92 +/- 0.08, implying a tight constraint of 18% - 28% on the variation of the stellar formation epoch along the sequence of spheroidal galaxies per decade of radius. The intrinsic dispersion of the relation, sigma([mu])((i))e = 0.40 +/- 0.03, does not vary with redshift, and the distribution of galaxy sizes, as well as the distribution in the plane of the effective parameters, does not vary among the clusters, as proven by the Kolmogorov-Smirnov tests. We conclude that whatever the mechanism driving galaxy evolution is, it does not significantly affect the properties of bright galaxies in the log r(e)[mu](e) plane at least since z = 0.64. The evolution of the zero point of the Kormendy relation is fully explained by the cosmological dimming in an expanding universe plus the passive luminosity evolution of stellar populations with high formation redshift (z(f) > 2).