1547845440-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_III__Chow_

3. GRADIENTS OF CONVEX FUNCTIONS ON RIEMANNIAN MANIFOLDS 435

DEFINITION H.32 (Convex functions (zero at the boundary) - <tXo (C)).

Let Cc M be a closed connected locally convex set with nonempty bound-

ary and interior. We denote by <tX 0 ( C) the collection of locally Lipschitz

convex functions f : C--+ JR satisfying flee = 0.

The derivatives of such functions have the following properties.

LEMMA H.33 (Convexity of D f (p) for f E ltXo ( C)). Let f E ltXo ( C)

and let p E 8C.

(i) Suppose l-1i E TpC n s;;-^1 is a sequence which converges to a point

v 00 in the boundary of T pC n s;;-^1 in s;;-^1. Then

(DvJ) (p)--+ 0.

Hence the Lipschitz extension D f (p) : TpC --+ JR satisfies

D f (p) (V) ~ Dv f (p) = 0 on TpC - TpC.

(ii) The function D f (p) : TpC --+ JR is convex. Hence by Proposition

H. 5, D f (p) is locally Lipschitz.

PROOF. (i) Since l-'i is in the interior tangent cone TpC, by (H.4) and

the definition of directional derivative, there is a sequence si --+ o+ such

that for alls E (0, si),

(H.22)

and

expP ( s l-'i) E int ( C)

f (expP (sl-'i)) - f (p) _ (DvJ) (p) :S 2 i

s '

Since Voo is in the boundary of TpC n s;;-^1 in s;;-1, there exist Si E (0, si)
such that

(H.23)
Consider the shortest path on s;;-^1 joining1.-1i to V 00 • By (H.22) and (H.23),
there is a point Vf on this path such that

expP (Si Vi') E 8C.

Let Lp be the Lipschitz constant off on some neighborhood of pin C. Since

f E ltXo (C), we have f (p) :== f (expP (si"\li')) = 0. Using this, we compute

IDv.J (p)I :S f (expP (sil-'i)) - f (p) + 2 _i

Si

f (expP (sil-'i)) - f (expP (si"\li')) +

2

_i

Si