- CONNECTED LOCALLY CONVEX SUBSETS IN RIEMANNIAN MANIFOLDS 421
a (0) = x, a (1) = y, and
L (a)= de (x, y).
Note that we may assume that
f3i ([O, 1]) c B (x, de (x, y) + 1) n C
for all i (the set on the RHS is compact). For every z EB (x, de (x, y) + l)nCthere exists Ez > 0 such that CnB (z, Ez) is convex. Consider the (relatively)
open cover{B (z, Ez) n B (x, de (x, y) + 1) n C} zEB(x,dc(x,y)+l)ne
of the compact metric space B (x, de (x, y) + 1) n C. By Lebesgue's number
lemma^5 , there exists 8 > 0 such that for every z EB (x, de (x, y) + 1) n C,B ( z, 28) C B (z', E z')
for some z' EB (x, de (x, y) + 1) nC. Hence C n B (z, 28) is convex for every
z EB (x, de (x, y) + 1) n C.
Since the lengths of the constant speed paths {f3i} have a uniform upper
bound, there exists m E N such that, for all 1 :::;; k :::;; m and for all i E N,
there exists Zk,i EB (x, de (x, y) + 1) n C such that
f3i ( [ k:
1
, ~]) c B (zk,i, 8).
Moreover, there exists a subsequence of i such that for any given k,
(1) the sequence Zk,i converges to some point Zk,oo as i-+ oo and
(2)for all k and i.
We then have(3". ([km-1' mk ]) c B (zk,oo, 28) n C
for all k and i. Since, for each k, the sequence {f3il[k-1 -"'--]}~ is uni-
m 'm i=l
formly bounded (in the ball B (zk,oo, 28)) and is equicontinuous because of
its constant speed assumption, it follows from the Arzela-Ascoli theorem
and a careful choice of subsequence ij that a subsequence of f3ii I [k-1 -"'--] will
m 'm
converge in c^0 to a c^0 path
ai[k-1 m 'm -"'--] in 13 (zk,oo, 28) n c
for each k = 1, ... , m. We now have constructed a limit c^0 path
a : [O, 1] -+ C.(^5) Recall that Lebesgue's number lemma says that if (X, d) is a. compact metric sr»ace
and if I!: is an open cover of X, then there exists li > 0 such that every ball of radius :::; li
is contained in a member of Q'..