1547845440-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_III__Chow_

422 H. CONVEX FUNCTIONS ON RIEMANNIAN MANIFOLDS

STEP 3 (Regularity of the path a). This follows from showing that a
satisfies L (a)= de (x,y). Given any partition {se}~ 0 of [O, 1], we have

N

L d (f3ij (sc-1), f3ij (sc)) :::; L (f3ij).

£=1

Taking the limit ij -t oo, we obtain

N

""d L.1 (a (sc-1), a (sc)):::; i-+oo ,lim L (f3i),

£=1

and hence

L (a) :::; de (x, y).

Since a is a curve in C joining x and y, we conclude

L(a) =de(x,y).

Given any s E (0, 1), there exists an open neighborhood U of a (s) in
M such that C n U is convex. In particular, for 0"1 < 0"2 sufficiently small,
we have a (s + O'i) EC n U for i = 1, 2. Since a has minimal length among
paths in C and since C n U is convex, we have

That is, a is locally length minimizing and hence a is a geodesic of ( M, g).

D

REMARK H.16. Alternatively, in the above proof, one should be able to
minimize length in a space of 'piecewise-short' broken geodesics a la Milnor
[129].

The following proposition gives a connection between connected closed
locally convex subsets and Aleksandrov spaces.

PROPOSITION H.17 (Locally convex subsets of Riemannian manifolds).

Let (Mn, g) be a complete Riemannian manifold with sectional curvature 2:

k. Let C be a connected closed locally convex subset in M. Then, as a length

subspace, C is an Aleksandrov space (with possibly nonempty boundary) with

curvature 2: k.

PROOF. Lemma H.15 implies that C is a complete locally compact length
space. Note that, from the definition of convex set and the Toponogov

comparison theorem for M, we know that C satisfies the local version of the

definition of Aleksandrov space with curvature 2: k. D