452 H. CONVEX FUNCTIONS ON RIEMANNIAN MANIFOLDS
PROOF. Let C ~ df:ia~~. sup Since r (x, P) is a broken geodesic, where
r (x, P)l[si-l,Bi] is a constant speed minimal geodesic joining Xi-1 to Xi, the
lemma follows from the claim that
de (xi-1, Xi) :::; C lsi - Si-1 I for i = 1, ... , m.
To see the claim, fix i and a point q E 1-^1 (!sup)· Let 'Y : [O, 1] -+ C
be the constant speed shortest path in C joining Xi-1 and q. Define q1 ~
'Y (/'-~i-.sup Bi-1^1 ) • It follows from the concavity off that
f(ql)2:'. fsup-Si f('Y(O))+ Si-Si-l j('Y(l)).
!sup - Si-1 fsup - Si-1
fsup - Si Si - Si-1 .r
= Si-1 + · Jsup
fsup - Si-1 · fsup - Si-1
=Si.By the definition of Xi = (xi-l)si-Bi-^1 , we have
de (Xi-1,Xi):::; de (Xi-1,q1).Hence the claim follows from
Si - Si-1 diam (C)
de (xi-1, q1) =de (xi-1, q) · f _. = f _ T (si - si-1).
sup Si-1 sup
D
A simple consequence of Lemma H.55 and the Arzela-Ascoli theorem is
COROLLARY H.56 (Limit r 00 of r (x, P) as IPI -+ 0). Let c, x, and f beas in Proposition H.53. Given a ·sequence of partitions Pk= {sk,i}~ 0 with
limk-+oo IPkl ~ o, there is a subsequence of broken geodesics r (x, pkj) which
converges as j -+ oo to a path
roo: [a,T]-+ C
uniformly on [a, T].
We have the following properties of r 00 •
LEMMA H.57. Let C, x, a, and f be as in Proposition H.53. Suppose TE
(a, !sup)· If Pk= {sk,i}:ko is a sequence of partitions with limk-+oo IPkl = 0
and such that as k -+ oo the broken geodesics r (x, Pk) converge to a path
r oo : [a, T] -+ C uniformly on [a, T], then ·
(1) r 00 (f (x)) = x,
(2) r 00 is Lipschitz,
(3) f (r oo (s)) = s for alls E [a, T],
(4)1. de(roo(so+cr),roo(so)) · 1
imsup < -----
o--+O+ er - JV'f(roo(so))J
for all so E [a, T).