- GRADIENTS OF CONVEX FUNCTIONS ON RIEMANNIAN MANIFOLDS 437
Note that \7 f (x, y) is not continuous in (x, y) (it takes eight discrete values).
Furthermore, l'V JI is also not continuous: l'V JI = 1 if lxl #- IYI, whereas
l'V fl=~ if lxl = IYI·.For functions in \!:Xo ( C), the generalized gradient is unique.LEMMA H.36 (Uniqueness of the generalized gradient). Let f E !:Xo ( C).
Then
(i) For any p E C, TpC is nonempty.(ii) If C is compact, then f S 0 on C. Consequently, unless f = 0 on
c,
finf ~inf {f (x) : x EC}< 0.(iii) For any p E C we can choose a generalized gradient \7 f (p) which
is contained in TpC.(iv) (Uniqueness) If finf < 0, then whenever f (p) > finf, we have
that (Dv J) (p) attains its negative minimum on TpC n s;-^1 at a
unique vector. Hence the generalized gradient \7 f (p) is unique and
nonzero.PROOF. (i) Suppose q EC and r > 0 are such that
(H.25) B (q, r) c C.
Since C is connected and locally convex, Lemma H.15 implies that there is a
smooth path/: [O, 1]-+ C, joining p = / (0) and q = / (1), which is minimal
among paths in C and which is a geodesic in M. Define
lo ~ inf inj ( / ( s)).
sE[O,l]By Lebesgue's number lemma, we can divide the geodesic I into (short)
segments 11 [si, 8 H 2 l, i :::;::: 0, ... , m - 2, where m ?: 2, such that
(1) each segment is a smooth minimal geodesic with length S i%-and
(2) 1\[si- 2 ,si] is contained in the convex set C n Ui, where Ui is an open
set, for i = 2, ... ,m.Here so = 0 and Sm = 1.
We now prove inductively that each/ (si), 1 Si Sm, is in int (C). Since
/ (sm) = q, by (H.25) we know that/ (sm) E int (C). Suppose
/ (si) E; B (t (si), ri) c int (C) nUi
for some i ?: 2 and ri > 0. We now prove/ (si-1) E int (C). Note that the
open 'geodesic cone'
{
O < s < sw and exp 7 (si_ 2 ) (sW)j[O,sw] is }
exp7(si-2) (sW) : the minimal geodesic joining/ (si-2)
and some point in B (t (si), ri)is contained in C n Ui. It is clear that the open geodesic cone contains