1547845440-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_III__Chow_

428 H. CONVEX FUNCTIONS ON RIEMANNIAN MANIFOLDS

DEFINITION H.22. Given a function f: S--+ IR and VE TpS, we define

the directional derivative of f in the direction V at p by

(H.8). ( Dv f) (p) ~ lim f ( expP ( s V)) - f (p) ,

s--+O+ S

provided the limit exists.

Let V be a cone in some finite-dimensional real vector space. A function

h : V --+ IR is positively homogenous of degree 1 if for every V E V and

>. E (0, oo) we have

h (.AV)= .Ah (V).

For example, if I· I is a norm on the vector space, then h (V) ~ IVI is

positively homogenous of degree 1.

Clearly we have

LEMMA H.23. For any VE TpS and r > 0, we have

Drvf = rDvf,

i.e., the function D f : V f--t Dv f is positively homogenous of degree l.

Let C c M be a connected locally convex set and let f : C --+ IR be

a convex function. For each p E C and s > 0 we define the difference

quotient function^9

(H.9)

f (expP (sV)) - f (p)

ls : V f--t --~----­

S

for any V E dom (ls) ~ {VE TpC: expP (sV) E int (C) }· Note that for

0 < s1 ::::; s2, we have dom ( ls 2 ) C dom (Js 1 ). Clearly

(H.10) (Dv f) (p) = lim ls (V).

s--+O+

REMARK H.24. Since expP ( s V) = p + s V on Euclidean space lEn, we

can rewrite the above difference quotient for C c IEn as

ls : V c--+ f (p + s V) - f (p).

s

The difference quotient and directional derivative have the following

properties.

LEMMA H.25 (Directional derivatives of convex functions). Let C c

(Mn, g) be a connected locally convex set in a connected complete Riemann-

ian manifold and let f : C --+ IR be a convex function.

(^9) 0f course, for any function defined in a neighborhood of a point pin a Riemannian
manifold, the difference quotient at p is well defined for s sufficiently small.