- BUSEMANN FUNCTIONS 305
r : [O, oo) -+ JRn is a ray with r (0) = 0 and d1/dt (0) = V, then the
associated Busemann function is easily computed using the law of cosines:
b'Y ( x) = S->00 lim ( d ( 0, 0 + s V) - d ( 0 + s V, x))
= s12.~ (s -Js^2 +Ix - 012 - 2s (x - o, v))
= (x - 0, V ).DEFINITION B.47. The Busemann function b : Mn-+ JR associated
to the point 0 E Mn is
b ~sup b'Y,
'Y
where the supremum is taken over all rays r emanating from 0.
This definition enables us to associated a Busemann function to a point.
In Euclidean space, the Busemann function associated to a point 0 E ]Rn is
just the distance function, because for all x E ]Rn,
b (x) = ( x -0, I~= ~I) = Ix - 01 = d (x, 0).
LEMMA B.48. The Busemann function b associated to 0 E Mn is
bounded above: for all x E Mn,
!"b(x)i :S d(x,O).
And b is uniformly Lipschitz with Lipschitz constant l: for all x, y E Mn,!"b(x) -b(y)I :S d(x,y).
PROOF. The first statement follows from (B.5). To prove the second,
note that for any x, y E Mn and c > 0, there exists a ray r emanating from
0 such that
b (x) - b (y) :S b'Y (x) + c - b (y) :S b'Y (x) + c - b'Y (y).
Hence by (B.6),
b (x) - b (y) :S d (x, y) + c.
D
The following result says that any sufficiently long minimal geodesic
segment emanating from a point 0 can be well approximated by a ray em-
anating from 0.LEMMA B.49. Given 0 E Mn, define e: [O, oo)-+ [O, 7r] by
e (r) = sup inf Lo (0-(0), p (0)),
a-ES(r) pER
where S (r) is the set of all minimal geodesic segments O" of length L ( O") ?: r
emanating from 0, and R is the set of rays emanating from 0. Then
lim e (r) = 0.
r->oo