A regular triangulation of a finite point set 
is a triangulation
of its convex hull obtained by lifting the points
to 
with assigned real heights and projecting the lower
faces of their convex hull.
The heights are chosen so that the projected cells are simplices.
Regular triangulations are also called coherent triangulations (De Loera et al.
2010).
For each fixed integer 
, Allamigeon et al. (2026) proved that the number
of regular triangulations of 
is 
as 
, where 
is the 
-dimensional simplex.
