Post-AGB binaries are surrounded by circumbinary disks of gas and dust that are similar to protoplanetary disks found around young stars. We aim to understand the structure of these disks and identify the physical phenomena at play in their very inner regions. We want to understand the disk-binary interaction and to further investigate the comparison with protoplanetary disks. We have conducted an interferometric snapshot survey of 23 post-AGB binaries in the near-infrared (H-band) using VLTI/PIONIER. We have fitted the multiwavelength visibilities and closure phases with purely geometrical models with an increasing complexity in order to retrieve the sizes, temperatures and flux ratios of the different components All sources are resolved and the different components contributing to the H-band flux are dissected. The environment of these targets is very complex: 13/23 targets need models with thirteen or more parameters to fit the data. We find that the inner disk rims follow and extend the size-luminosity relation established for disks around young stars with an offset toward larger sizes. The measured temperature of the near-infrared circumstellar emission of post-AGB binaries is lower (Tsub~1200K) than for young stars, probably due to a different dust mineralogy and/or gas density in the dust sublimation region. The dusty inner rims of the circumbinary disks around post-AGB binaries are ruled by dust sublimation physics. Additionally, a significant amount of the circumstellar H-band flux is over-resolved (14 targets have more than 10% of their non-stellar flux over-resolved) hinting for more structure from a yet unknown origin (disk structure or outflow). The amount of over-resolved flux is larger than around young stars. Due to the complexity of these targets, interferometric imaging is a necessary tool to reveal the interacting inner regions in a model-independent way.
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