The aims of this article are to infer the radial and vertical structure at ~au scale of these disks, and the properties of the dust grains: composition and characteristic size. We find that dust at the inner rim of the disk has a sublimation temperature Tsub~1800K. A ring structure is confirmed for approximately half the resolved objects; these rings are wide {delta}r/r>=0.5. Such wide rings exclude an inner rim shaped as a vertical wall, or with a rounded shape, and seem to favor an inner rim with a wedge-like cross-section. The inner disk 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. We find in some of the objects a halo component, fully resolved at the shortest interferometer spacing, that is related to the HAeBe class.

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