436 H. CONVEX FUNCTIONS ON RIEMANNIAN MANIFOLDS
By the Toponogov comparison theorem (the sectional curvature is bounded
from below on compact subsets of M), there is a constant Co < oo such
that for fVI , IWI :S 1 we have
Hence
d (expP (V), expP (W)) :S Co IV - WI.
IDvJ (p)/ ::::; Lp Co lsi Vi - Si Vi'/ + 2 -i
Si
= Lp /Vi -Vi'/+ 2-i-+ 0 as i-+ oo.(ii) Since TpC is convex and hence connected, we may apply Lemma
H.26(ii) to obtain that DJ (p): TpC-+ IR is convex. That its Lipschitz ex-
tension D f (p) : T pC -+ IR is convex follows from a simple limiting argument.
The lemma is proved. D
The following is a simple nonsmooth example.EXAMPLE H.34 (Square in the plane). LetC ~ [-1, 1] x {-1, 1] c IR^2 ,which is a compact convex subset. Define f: C-+ IR by
f (x, y) = -d ((x, y), 8C).
Then
(H.24) f (x, y) =max {/xi, /YI} - 1
and f E <!:xo (C), i.e., f is convex and !lac = O. Let
D ~ {(x, y) EC: lxl = IYI} ·
Then f is smooth in C - D and only Lipschitz continuous on D.
The above example has the following generalized gradient.EXERCISE H.35. Show that (or at least convince yourself that) in Ex-
ample H.34:
(1)(2)
· (a) If x > IYI, then \7 f (x, y) = (-1, 0).
(b) If x < - I y I, then V' f ( x, y) = ( 1, 0).
(c) If y > /x/, then \7 f (x, y) = (0, -1).
(d) If y < - lxl, then 'V f (x, y) = (0, 1).(a) If x = y > 0, then \7 f ( x, y )" = ( -~, -~).
(b) If x = y < 0, then \7 f ( x, y) = ( ~, ~).(c) If x = -y > 0, then \Jf (x,y) = (-~, ~).
(d) If x = -y < 0, then \7 f (x, y) = (~, -~).