- WEAK SOLUTION FORMULATION 371
then there exists a C^2 function 'I/; (x) defined on Bp (2r) c TpM
such that (f o exp_;^1 +'I/;) ( x) is a concave function on Bp ( r) with
respect to the Euclidean metric g (p). In particular, f is locally
Lipschitz on B (p, r).PROOF. (i) We may assume C + c > 0 since otherwise we are done.
Since \7\7 [d(x,p)^2 ] (p) = 2g(p), where g is the metric, by the continuityof \7\7 [ d ( x, p)^2 J near p, there. exists r > 0 (depending only on g and how
big of a ball centered at p fits inside W) such that \7\7 [d (x;p)^2 ] ?; g in
B (p, r) C W Let 'ljJ ( x) = - ( C + c.) d ( x, p)^2. Then
Hess supp ('I/;) = \7\7 'I/; :::; - ( C + c) g.Since Hess supp (!) :::; C, we conclude Hess supp (f + 'I/;) :::; -c on B (p, r).
Note that we may also assume [\7'1/;[ :::; C' on B (p, r) for some constant
C' < oo.
(ii) Let ry : [O, a] --> W be any geodesic parametrized by arc length. Toverify that -f is a convex function on (W, g) in the Riemannian sense, we
need to show that f o ry : [O, a] -->JR. is a concave function. Since f o 'Y also
satisfies Hesssupp (f o ry) :::; 0 on [O, a], this reduces the original problem to a
1-dimensional problem.
For every point so E (0, a) and c: > 0 there exists a C^2 function cp (s)
defined on a subinterval (so - 8, so + 8) such thatf o ry (so)= <p (so) and f o ry (s) :::; <p (s)
for alls E (so - 8, so+ 8), and cp" (so) :::; c:. Let
0 (s) ~ <p (so)+ cp' (so) (s - so)+ c: (s - so)^2.
Note that tjJ (so)= <p (so), <p' (so)= cp' (so), and 0" (s) = 2c: > c: 2: cp" (so).
Hence there exists 81 E (0, 8) such that
f o ry (so)= 0 (so) and f o ry (s) < 0 (s)
for s. E (so - 81,so + 81) - {so}. We claim that f o ry(s) :::; <p(s) for all
s E [O, a]. By taking c: --> 0, the claim then implies
f o ry (s) :::; f o ry (so)+ cp' (so) (s - so) for all so, s E [O, a].
We conclude f o ry ( s) is concave on [O, a].
Finally, suppose the claim is false; then there exists s1 E [O, a] - {so}
such that f o ry (s1) = tjJ (s1). Suppose s2 E (so, s1) is a minimum point of
f o ry (s) - tjJ (s) on [so, s1]. Then
(7.143) f o ry (s) 2: f o ry (s2) + [cp' (so)+ 2c: (s2 - so)] (s - s2) + c: (s - s2)^2
on [so, s 1 ]. On the other hand, by our hypothesis on f o ry, there exists a C^2
function cp 2 ( s) defined for s near s2 such that
f o 'Y (s) :::; f o ry (s2) + <p; (s2) (s - s2) + ~ (s - s2)^2 ,