Sentence:  Regular <WSD>triangulations</WSD> are a class in-between lifting and lexicographic <WSD>triangulations</WSD>. Their main draw-back in the context of this paper is that regularity depends on the specific realization and not only on the oriented matroid. 
Meaning of the word marked with <WSD> tags:  A subdivision of a planar object into triangles, and by extension the subdivision of a higher-dimension geometric object into simplices.