432 H. CONVEX FUNCTIONS ON RIEMANNIAN MANIFOLDS
From Lemma H.25(i), we have for i 2: io
(DvJ)(pi) ::::; f ( expPi (so Vi)) -_ f (Pi) ::::; (Dv=f)(poo) + 3c.
soThis proves the upper semi-continuity of D f. D
A function </> : ]Rn --+ JR is said to be (finite) sublinear if </> is convex
and positively homogeneous of degree 1.
We have the following.
LEMMA H.27 (Sublinearity of directional derivative function). Let C C
M be a connected locally convex set and let f : C --+ JR be a convex function.
(i) If for some p E C and for some c > 0 and L < oo we have
.jf (q1) - f (q2)I :SL d (q1, q2) for all qi, q2 EB (p, c) n C, then
I Dw f (p) - Dv f (p) I :S L I W - VI
for all V, W E TpC. Hence, given p E int ( C), the directional deriv-
ative functionV f---'t Dv f (p)
is Lipschitz continuous.
(ii) The function V H-D f (p) (V) is sub linear for p E int ( C). Moreprecisely, if f is Lipschitz in some neighborhood of p E C, then the
function V r-+ D f (p) (V). is sublinear.
PROOF. (i) We computeIDw f (p) - Dv f (P)I
. f (expP (sW)) - f (p). f.(expP (sV)) - f (p)
hm - hm
s-to+ s. s-to+ s
. f (expP (sW)) - f (expP (sV))
hm
s-tO+ S. Ld ( expP (sW), expP (sV))
< hm
- s-tO+ S
=LIW-Vj.
(ii) The sublinearity of Dv f (x) in V follows from Lemma H.23 and
Lemma H.26(ii). The lemma is proved. D
The following is an easy consequence of Lemma H.25(iii); for the defini-
tion of the directional derivative of a concave function, see (H.31) below.
EXERCISE H.28 (Directional derivative of concave functions). Show thatif f : C --+ JR is concave, where C c M is a connected locally convex set,
and if p E int (C), then (Dv f) (p) exists for all V E TpM. Moreover, V H-
(Dv f) (p) is a concave function which is positively homogenous of degree 1
and