Context. The formation and the evolution of protoplanetary disks are important stages in the lifetime of stars. Terrestrial planets form or migrate within the innermost regions of these protoplanetary disks and so, the processes of disk evolution and planet formation are intrinsically linked. Studies of the dust distribution, composition, and evolution of these regions are crucial to understanding planet formation.; Aims. We built a homogeneous observational dataset of Herbig Ae/Be disks with the aim of spatially resolving the sub au-scale region to gain a statistical understanding of their morphological and compositional properties, in addition to looking for correlations with stellar parameters, such as luminosity, mass, and age.; Methods. We observed 27 Herbig Ae/Be stars with the GRAVITY instrument installed at the combined focus of the Very Large Telescope Interferometer (VLTI) and operating in the near-infrared K-band, focused on the K-band thermal continuum, which corresponds to stellar flux reprocessed by the dust grains. Our sample covers a large range of effective temperatures, luminosities, masses, and ages for the intermediate-mass star population. The circumstellar disks in our sample also cover a range of various properties in terms of reprocessed flux, flared or flat morphology, and gaps. We developed semi-physical geometrical models to fit our interferometric data.; Results. Our best-fit models correspond to smooth and wide rings that support previous findings in the H-band, implying that wedge-shaped rims at the dust sublimation edge are favored. The measured closure phases are generally non-null with a median value of ~10°, indicating spatial asymmetries of the intensity distributions. Multi-size grain populations could explain the closure phase ranges below 20–25° but other scenarios should be invoked to explain the largest ones. Our measurements extend the Radius-Luminosity relation to ~104L⊙ luminosity values and confirm the significant spread around the mean relation observed by PIONIER in the H-band. Gapped sources exhibit a large N-to-K band size ratio and large values of this ratio are only observed for the members of our sample that would be older than 1 Ma, less massive, and with lower luminosity. In the mass range of 2 M⊙, we do observe a correlation in the increase of the relative age with the transition from group II to group I, and an increase of the N-to-K size ratio. However, the size of the current sample does not yet permit us to invoke a clear, universal evolution mechanism across the Herbig Ae/Be mass range. The measured locations of the K-band emission in our sample suggest that these disks might be structured by forming young planets, rather than by depletion due to EUV, FUV, and X-ray photo-evaporation.
