440 H. CONVEX FUNCTIONS ON RIEMANNIAN MANIFOLDSfor any A. E (0, oo ). By the definition of the generalized gradient, we have(D\lf(p)+AV f) (p) = IV' f (p) +A. VI (D 'Vf(p)+>-v !) (p)
IY'f(P)+WI2:: - IV'! (p) +A.VI· l\7f (p)I
= - IV'! (p)I VIV'! (P)l
2
+ 2A. (V'f (p) 'V) + 0 (A.2)
= - IV'f (P)l^2 - A. (\i'f (p), V) + 0 (A.^2 ).Since DJ (p): TpC-+ IR is convex (which follows from Lemma H.33(ii)), we
have(D\lf(p)+>.v f) (p) = (1 +A.) (D (1+) 1 \lff~)+->--vi) V' (l+) (p)
::::: (1 +A.) Cl~ A.) D\lf(y)f (p) + (l ~A.) (Dv f) (p))
= -l\i'f(p)l^2 +A.(Dvf)(p).
We have proved for any A. > 0,
- (V' f (p) , V) + 0 (A.) ::;: ( Dv f) (p).
Inequality (H.28) follows by taking A. -+ 0. D
In general, if J : M -+ IR is a C^1 function, then wherever V' f =/= 0, the
level sets { x E M : f ( x) = s} of f are C^1 hypersurfaces and the vector field
V' f is orthogonal to these level sets. Although convex functions are not
necessarily C^1 , we have the following.^10LEMMA H.38 (V' f makes acute angles with 'f-nonincreasing' geodesics).
Let f E <!::to ( C). Suppose that !inf < 0. If x, y E C are such that f ( x) > finf,
f (x) 2:: f (y), and x =/= y, then for any constant speed geodesic/: [O, 1]-+ C
with/ (0) = x and/ (1) = y, we have.L (\i' f ( x) , 'Y ( O)) ::;: ~.
PROOF. Clearly 'Y (0) E TpC, wh1ch, by Lemma H.37, implies- (V' f (x), 'Y (0)) ::;: (D· o f) (x) = lim f (! (s)) - f (x).
!'( ) s--+O+ SOn the other hand, since f is convex,
f (! (s)) ::;: (1 - s) f (! (0)) +sf(! (1))
= (1 - s) f (x) +sf (y)
::;: f (x)
for s E [O, 1]. We conclude that (V' f (x) , 'Y (0)) 2:: O. D
(^10) The geodesic'"'( joins a point at a higher 'f-level' to a point at a lower f-level; this
is what we mean by 'Y being 'f-nonincreasing'.