Barycentric subdivision is a recursive refinement of a simplex using the geometric
centroids, or barycenters, of its nonempty faces and
of the simplex itself (Hatcher 2002, pp. 119-120).
For a simplex with vertices 
,
..., 
in Euclidean space, its barycenter is
 |
A 0-simplex is unchanged. For 
, first subdivide each facet recursively. For every 
-simplex in these subdivisions,
take the convex hull of its vertices
together with 
.
These are the 
-simplices of the barycentric subdivision. In two dimensions,
the construction divides a triangle into six triangles
using its side midpoints and geometric
centroid.
For an 
-simplex 
with 
, every simplex 
in its barycentric subdivision satisfies
 |
where 
denotes the generalized diameter measured
using the Euclidean metric (Hatcher 2002, pp. 120-121).
After 
iterations, the largest generalized
diameter is at most 
, which tends to zero as 
.
