310 B. SOME RESULTS IN COMPARISON GEOMETRY
PROOF. We prove both inclusions s;:: and 2 to obtain equality.
(s;::) If x E Iffi'°Y•' there exists r > 0 such that d(x,1(r+s)) < r. We may
assume r > t - s, because d ( x , r ( q + s)) < q for all q ?: r. Then we have
d (x, B (r (s + r), r - (t - s))) < r - (r - (t - s)) = t - s.
This proves that d ( x, Iffi'"Yt) < t - s, because
B (r ( s + r) , r - ( t - s)) = B ( t + ( s + r - t) , s + r - t) s;:: Iffi'"Yt.
(2) If d (x, Iffi'"Yt) < t -s for some x E Mn, then there exists y E Iffi'"Yt such
that d(x,y) < t - s. By definition of Iffi'"Yti this implies there is r E (O,oo)
such that y E B (rt ( r) , r). Thus we have
d(x,1(s+(t-s+r))) =d(x,1(t+r))
:Sd(x,y)+d(y,1(t+r)) <t-s+r,
and hence x E B ( ')' ( s + ( t - s + r)) , t - s + r) s;:; Iffi'"Ys. 0
DEFINITION B.58. Given a point 0 E Mn ands E [O, oo), the sublevel
set of the Busemann function associated to 0 is
Cs~ n Ry.,
)'ER(O)
where R (0) is the set of all rays emanating from 0.
Note that
Cs= Mn\ U Iffi'"Ys"
)'ER(O)
The next several results explain both the name and the usefulness of Cs.
LEMMA B.59. For every choice of origin 0 E Mn ands E [O, oo), the
sublevel set of the Busemann function associated to 0 is given by
Cs= {XE Mn : b (x) :SS},
where b is the Busemann function associated to 0. In particular, we have
Cs s;:: Ct
whenever 0 :S s :St, and
8Cs = { x E Mn: b (x) = s}
because b is continuous.
PROOF. Observe that b'"Ys = b'"Y - s, since for all x E Mn,
b'"Ys (x) ~ t-->oo lim b'"Ys,t (x) = t-->oo lim (t - d (rs (t), x))
= - s + lim ( s + t - d (r ( s + t) , x)) = -s + b'"Y ( x).
t-->oo