450 H. CONVEX FUNCTIONS ON RIEMANNIAN MANIFOLDS
in particular, if IV f (Yo) I > c, then
dc(y,ys) :S IVJ(:o)l-c.
PROOF. By LemmaH.36(iv) and f (yo) <!sup it follows that IV f (Yo)I-/=
o. Let Wo ~ IY'J{yo)I V f (Yo). Then
Wo E TyoC n s;o-l and Dwof (Yo)= IV f (yo)I.Choose an c1 > 0 such that
- 1
1
(IV f (yo)I - c1) 2:: IV f (yo)I - c.
+c1We identify a neighborhood of (yo, Wo) in TM with Uo x Ty 0 M, where Uo
is a neighborhood of y 0 , U 0 is compact, and Uo n C is convex. By the lower
semi-continuity of DJ: LJpECTpC ~JR at (yo, Wo) (Lemma H.26(iii)), there
is a neighborhood U1 c U1 c Uo of Yo and a neighborhood V1 in Ty 0 M of
W 0 satisfying V 1 c TyC for all y E U1 n C (by a natural identification, TyC
may be considered as a subset of Ty 0 M) such that
Dw f (y) 2:: Dw 0 f (yo) - c1 =IV f (yo)I - c1 and IWI < 1 + c1
for all (y, W) E (U1 x V1) n (LJpECTpc).
Let W : U1 ~ V1 be a continuous map with W (Yo) = Wo. Defineh: U1 nC ~ [O,oo) by
_,__ {. expy(sW(y)) EU1nC forsE(O,s1],}
h(y) --,... sup s1. d ( ~ ).ds expy sW (y) E V1 for s E (0, s1]
Clearly h(y) > 0 and h is lower semi-continuous on U 1 n C. Choose a
neighborhood U2 c U2 c U1 of yo; we defines* ~inf {h (y), y E U2 n C}.
Note thats* > 0. Given a y E U2 n C, we define a concave function
g:[O,!s*J~lRby g ( s) ~ f ( expy ( s W (y))). By the choice of U2, U1, and V 1 , we have9~ (s) = D -Js expy( sW(y) )f (y) 2:: IV f (yo) I - c1;
hence for s E [o, !s*]
f ( expy ( sW (y))) - f (y) =las g~ (s) ds 2:: s (IV f (Yo)I - c1).
This implies that expy (sW (y)) E C/(y)+s(IV'f(yo)l-c-i)· On the other
hand, by the choice of V1 and W, we have