420 H. CONVEX FUNCTIONS ON RIEMANNIAN MANIFOLDS
LEMMA H.15 (Existence of minimal geodesics in C between points). If C
is a connected closed locally convex subset of a complete Riemannian man-ifold (Mn,g), then for every x,y E C there exists a smooth geodesic (of
(M,g)) a: [O, 1]---+ C with a (0) = x, a (1) = y, and
L (a)= de (x, y) ~inf L (,8),
(3
where the infimum is taken over all paths ,6 : [O, 1] ---+ C with ,6 (0) = x and
,e (1) = y.
PROOF. We prove the lemma in three steps.
STEP 1 (Existence of shorter piecewise C^00 geodesic paths). For every
x, y EC and any path '/J : [O, 1] ---+ C with '/J (0) = x and '/J (1) = y, we claim
that there exists a piecewise C^00 geodesic path
,6: [0,1]---+Cwith ,6 (0) = x, ,6 (1) = y, and
L (,8) ::; L ('/J).
To see this claim, we observe that, since ,6 is continuous, there is an ro E
(0, oo) such that '/J ([O, 1]) c B (x, ro). For every z E B (x, ro) n C thereexists Ez > 0 such that CnB (z,Ez) is convex. Since UsE[O,l]B ('/J (s) ,E,B(s))
is an open cover of the compact set '/J ([O, 1]), it is easy to see that there is
a partition so = 0 < s1 < · · · < Sm = 1 such that'/J ([sk-1, sk]) c B ('/J (sk), E,B(sk))
for some Bk E (sk-I, sk), k = 1, ... , m.
We define
,6 / [sk-1,sk]
to be the minimal geodesic joining '/J (sk-I) and '/J (sk)· It is easy to verify
that the broken geodesic ,6 has all the properties in the claim. Clearly we
also haveL (,8) < oo and de (x, y) < oo.
STEP 2 (Existence of minimizing path a). Let {'/Ji}. be a length
iEN
minimizing sequence for inf(3 L (,8) = de (x, y). For each i E N, let ,Bi :
[O, 1] ---+ C be a piecewise C^00 path constructed from '/Ji as in Step 1. If
necessary, we reparametrize so that each ,Bi has constant speed. Then the
piecewise C^00 paths {,Bi} iEN form a length minimizing sequence_lim L (,Bi)= inf L (,8) =de (x, y).
i-+oo (3
As typical in the calculus of variations, we shall use the direct method to
prove that a subsequence of {,8ihEN converges to a path a : [O, 1] ---+ C with