§ Barycentric subdivision: edge length decreases
For a edge , subdiving the edges into two at the center produces two edges
both the original length. Given a triangle , we wish to prove that
subdividing the triangle by joining the barycenter to the vertices reduces edge
length by of the maximum length. More generally, we wish to show that the
edge length decreases to of the largest length for an dimensional
The barycenter is at the location . The distance
from a vertex is .
By Cauchy Schwarz, we have that . One of the terms,
where will be zero, and the other (n-1) terms are at most , the length of the longest edge.
This gives , hence the edge length decreases by a factor of .