- GRADIENTS OF CONVEX FUNCTIONS ON RIEMANNIAN MANIFOLDS 439
Then I! Vi +! Vz I < 1,! Vi +! Vz E T pC, and! Vi +! Vz -/= 0 by Lemma
H.25(iii). From the convexity of (Dv f) (p) in V, we have
(n~v 1 +~v 2 f) (p) :S ~ (DVif) (p) + ~ (Dv 2 J) (p) = _min (Dvf) (p).
VETpCns;-^1It .c 10 11 ows f rom l~Vi+~Vil ~Vi+~v^2 E T-p C n sn-l P th a t
This is a contradiction and hence the minimum in (H.26) is attained by a
unique vector in TpC n s;-^1. This proves (iv) and the lemma. 03.2. Properties of generalized gradients of convex functions.
Recall that C:Xo ( C) is given by Definition H.32 andfinf ~inf {f (q) : q EC}.
LEMMA H.37 (Directional derivatives and the generalized gradient). Letf E C:Xo (C). Suppose finf < 0.
(1) If p EC is such that f (p) = finf, then
(H.27) (Dv f) (p) 2: - (\/ f (p), V) for any VE TpM,
where ! f (p) is any generalized gradient vector.
(2) If p E C is such that f (p) > finf, then
(H.28) (Dv f) (p) 2: - (\/ f (p), V) for any VE TpC,
where ! f (p) is the unique generalized gradient vector.
PROOF. (1) If f (p) = finf, then p E int (C) and
(Dv !) (p) 2: 0
for all VE TpM· Hence, by (H.18), we have
I\/ f (P)I =mm. { ( Dv !VI f) (p) : VE TpM - { .... 0 } }.
Thus, for all V E TpM,(Dv f) (p) 2: !VI· I\/ f (P)I 2: - (\/ f (p), V) ·
(2) If f (p) > finf, then there is a (unique) ! f (p) E TpC by Lemma
H.36(iii). For any VE TpC, by Lemma H.18(ii) we have
! f (p) + .\ v E TpC