460 I. ASYMPTOTIC CONES AND SHARAFUTDINOV RETRACTION
LEMMA I.8 (Sublevel set of bp)· For every choice of basepoint p E Mn
and s E [O, oo), the sublevel set of the Busemann function associated to p is
given byMoreover,
8C 8 (p) = {x E Mn: bp (x) = s}and if 0 ::::; s ::::; t, then
Cs(P)CCt(p).
Since bp (x) ::::; d (x,p), the sublevel sets of the Busemann function com-
pare to geodesic balls in the following way (for a proof, see Qorollary B.60
of Volume One).. COROLLARY I.9 (Sublevel set of bp contains geodesic ball). For any p E
M and s > 0 we have
B (p, s) c Cs (p).
1.1.3. Convex and totally convex subsets.
Recall that we defined convexity and local convexity for subsets in Rie-
mannian manifolds in subsection 2.1.2 of Appendix H. We have a few vari-
ants on the notion of convex set.(1) A subset E c Mn is said to be totally convex if for all x, y E E
every geodesic 'Y (not necessarily minimizing) joining x and y is
contained in E.^1(2) A subset E c Mn is said to be strongly convex if for any x, y E f;
the minimal geodesic f3 from x to y is unique and its interior int (/3)
is contained in E.
As we shall see, the above notions are particularly useful for complete
noncompact manifolds with nonnegative sectional curvature. We first give
some examples of such subsets. In the compact case we have the following.
EXAMPLE I.10 (Round sphere). Let Mn = sn be the unit n-sphere,
which we may think of as the subset
sn = { x E JRn+l : lxl = 1} ·
(1) A nonempty subset of sn is totally convex if and only if it is the
whole sn.
(2) For any a E (0, l] the subset Ea ~ {x E sn: Xn >a} is strongly
convex.
(3) The upper hemisphere Eo = {x E sn: x'f!-> O} is not strongly con-
vex.The following is a trivial noncompact example.
EXAMPLE I.11. A convex set in Euclidean space is totally convex.(^1) See Definition 8.1 on p. 134 of [30]. Note that in Definition B.53 on p. 307 of Volume
One the hypothesis that the geodesic a is minimizing should be removed.