438 H. CONVEX FUNCTIONS ON RIEMANNIAN MANIFOLDSLet U 1 be an open neighborhood of I' ( s1) such that C n U1 is a convex
set. Now B (!' (s1), r1) c int (C) nU1 for some r1 > 0 and the open cone
{
0 < s < sw and exp1'(o) (sW) I [O,sw] is }
exp1'(0) (sW): the minimal geodesic joining I' (0)
and some point in B (!' (s1), r1)contains every point I' ( s), 0 < s < s 1. Hence ')'^1 ( 0) E TpC and therefore TpC
is nonempty.
(ii) We prove f :S 0 on C by contradiction. Suppose that this statement
is false. Since C is compact and f is continuous on C and vanishes on the
boundary, there is a qsup E int ( C) such that
f (qsup) =sup f (q) > 0.
qEC
Consider any minimal geodesic segmentI' : [-10, 10] -+int (C)
with I' (0) = qsup· Then f o I' is a convex function on [-10, c] which attains
its maximum at s = 0. Hence f o I' is a constant function and therefore f is
a constant function in some neighborhood of qsup· Since we can prove that
f is a locally constant function at any point q with f (q) = supqEC f (q) andsince C is connected, we conclude that f is a constant function on C with
f (qsup) > 0. This is impossible because f = 0 on 8C. The claim is proved.
(iii) We only need to prove (iii) for p E 8C. By (ii) we have (Dv f) (p) :S 0for all VE TpC. So (iii) now follows from (i) and Dv f (p) = 0 on TpC -TpC
by Lemma H.33(i).
(iv) For p E 8C, let I' : [O, 1] -+ C be a geodesic which is minimal in Camong those paths connecting p to some point q E int (C) with f (q) < 0.
By the proof of (i) we have
')'^1 (0) E TpC.By the convexity off o I' (s), by f o I' (0) = 0, and by f o I' (1) < 0, we
conclude that ·
D1''(o)f (p) < 0.
On the other hand, for p E int ( C), if ( Dv f) (p) has a nonnegative mini-mum on TpCns;-^1 = s;-1, then from Lemma H.31 we conclude that f has
a local minimum at p. The convexity of f implies that p must be a global
minimum point of f, which contradicts f (p) > !inf. Hence
(H.26) min { (Dv f) (p): v E TpC n s;-^1 } < o.
Now let p EC and suppose that there are two distinct vectors Vi and 112
in TpC n s;-^1 such that
(Dvif) (p) = (Dv 2 f) (p) = _!!!in (Dv f) (p) < O.
VETpCns;-^1