Chew conjectured that the Delaunay triangulation
is a t-spanner [Che89] for some constant t.
Dobkin et al. [DFS90] established this
for
t = π(1 + )/2 5.08.
The value of t was improved to
2π/(3 cos(π/6)) 2.42
by Keil and Gutwin [KG92],
and further strengthened in [BM04].
Chew showed that t is
π/2 1.57 for points on
a circle, providing a lower bound.
``It is widely believed that, for every set of points
in
^{2}, the Delaunay triangulation is a
(π/2)-spanner'' [NS07, p. 470].
This history suggests the special case posed above.
There is a new forthcoming result: [CKX09].