Context. It is now generally accepted that the near-infrared excess of Herbig AeBe stars originates in the dust of a circumstellar disk. Aims. The aims of this article are to infer the radial and vertical structure of these disks at scales of order one au, and the properties of the dust grains. Methods. The program objects (51 in total) were observed with the H-band (1.6micron) PIONIER/VLTI interferometer. The largest baselines allowed us to resolve (at least partially) structures of a few tenths of an au at typical distances of a few hundred parsecs. Dedicated UBVRIJHK photometric measurements were also obtained. Spectral and 2D geometrical parameters are extracted via fits of a few simple models: ellipsoids and broadened rings with azimuthal modulation. Model bias is mitigated by parallel fits of physical disk models. Sample statistics were evaluated against similar statistics for the physical disk models to infer properties of the sample objects as a group. Results. We find that dust at the inner rim of the disk has a sublimation temperature Tsub~1800K. A ring morphology is confirmed for approximately half the resolved objects; these rings are wide delta_r>=0.5. A wide ring favors a rim that, on the star-facing side, looks more like a knife edge than a doughnut. The data are also compatible with a the combination of a narrow ring and an inner disk of unspecified nature inside the dust sublimation radius. The disk inner part has a thickness z/r~0.2, flaring to z/r~0.5 in the outer part. We confirm the known luminosity-radius relation; a simple physical model is consistent with both the mean luminosity-radius relation and the ring relative width; however, a significant spread around the mean relation is present. In some of the objects we find a halo component, fully resolved at the shortest interferometer spacing, that is related to the HAeBe class.
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