  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.
