1547845440-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_III__Chow_

2. CONNECTED LOCALLY CONVEX SUBSETS IN RIEMANNIAN MANIFOLDS 427

Geometrically it is clear that there exists r 0 E (0, r) such that^8

B (p, ro) c int(~).

Hence we have proved that f is locally bounded from above on B (p, ro);

Step 1 is finished.

Step 2. f is locally bounded from below in int (C). For any p E int (C),

let ro be as in Step 1 so that f on B (p, ro) is bounded from above by a

positive constant Co. Now we shall show that f is bounded. from below on

B (p, ro/2). For any point q EB (p, r 0 /2), let

s I---+ expP ( s V) ,

with IVlg(p) = 1, be the minimal geodesic from p to q, defined for 0 s s s sq.

If f (q) 2': 0, then we are done. On the other hand, if f (q) s 0, then we

have

f (p) S ro f ( expP (Sq V)) + sq f ( expP (-ro V))

ro +Sq ro +Sq

2

s3f(q)+Co.

Hence f is bounded from below by min{£ (J (p) - C 0 ), O} on B (p, r 0 /2).
Step 3. f is locally Lipschitz in int (C). For any p E int (C), let ro be as
in Steps 1 and 2 such that Iii on B (p, ro/2) is bounded by a constant Ci.
For any qi and q2 in B (p, ro/4) with f (qi) Sf (q2), let/ (s) be the minimal
unit speed geodesic joining qi = / (0) and q2 =/(so). We extend/ further,

from the end q2, until it intersects 8B (p, ro/2) at some point <fa ~ / (si).

It is clear that so < si, so S ro/2, and si 2': ro/4. Using the convexity of

f o / (s), we compute

f(q2)=fo1(so)S (i-s

0

)J(/(O))+s^0 Jo1(si),

s1 s1

f ( q2) - f ( q1) < f ( <12) - f ( q1) < 8C1.

so - s1 - ro

Hence we proved that f is Lipschitz in B (p, ro/4) with Lipschitz constant

801

ro

As a simple consequence of its locally Lipschitz property, f is continuous

in int (C). 0

2.2.3. Directional derivatives of convex functions.
Given a subset S c M, recall that its interior tangent cone, as defined
in (H.4), is denoted by TpS.

(^8) Here we used the condition that (B (p, 2r) , g) is 100 ~n 2 -close to the Euclidean ball
(B (2r), gEuc) in the 02 -Cheeger-:-Gromov topology.