1547845440-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_III__Chow_

2. CONNECTED LOCALLY CONVEX SUBSETS IN RIEMANNIAN MANIFOLDS 429

(i) For p EC and 0 < s1 :::; s 2 , we have

(H.11)

for all V E dom ( J 32 ). Hence the directional derivative function on

TpC,

V r+ Dvf (p) E [-oo, oo),

as defined in (H.8), exists for all p EC.

(ii) If f is locally Lipschitz at p E C in the sense that there exist c > 0

and L E [O, oo) such that

(H.12)

(H.13)

If ( q) - f (p) I :::; Ld ( q' p)

for q EB (p, c) n C, then for any s > 0 with ls VI < c we have

IJs (V)I :::; LIVI

for all V E dom ( J 3 ) and we have

IDv f (p)I:::; LIVI

for any VE TpC.

(iii) For any V E TpC, if -V E TpC, then

(H.14)

(H.15)

- (D-v J) (p) :::; (Dv f) (p).

In particular, if (Dvf) (p) < 0, then (D-vf) (p) > 0. Hence, if

p E int ( C), then we have

max { ( Dv J) (p) : V E s;-^1 } ~ 0,

where s;-^1 c TpM denotes the sphere of unit vectors.

PROOF. (i) Lett= !~ E (0, 1]. Inequality (H.11) follows from

f (expP (s1 V)) - f (p)

s1

f ( expP ( ( 1 -t) 0 + t ( s2 V))) -f (p)

s1

(1-t)f (expPo) +tf (expp(s2V))-f (p)

< ~---'-~----'--~~~

S1

f ( expP ( s2 V)) - f (p)

s2

where we used that f is convex. From this monotonicity formula and (H.10)

we conclude the existence of Dv f (p).

Note that for p E&C, the value of Dv f (p) could be -oo. For example,

f (x) = -ft has direCtional derivative -oo at 0 E [O, 1] ~ C. However if

p E int ( C), then Dv f (p) is finite for all V E TpC. ·