Suppose that instead of the usual circle, we pick some other convex set D and make the Delaunay triangulation of a finite planar point set with respect to this set, i.e. connect two points if there is a homothet of D that contains both of them on its boundary and no points inside. If the points are in a general position such that no four fall on the boundary of a homothet of D, then we still obtain a triangulation (plus some infinite face). Can anyone provide a good reference for this statement, how should I cite it? I need it not only for smooth D but also for polygons.

Perhaps the earliest reference is this:
This more recent dissertation at the FernUniversität Hagen may be more directly useful:
I extracted this figure from her thesis, p.58:
Google Scholar finds 22 papers post2000 that cite this dissertation. 

