- ELEMENTARY ASPECTS OF CONVEX ANALYSIS ON EUCLIDEAN SPACE 415
where f' is the derivative function defined a. e. on [a, b]. (See Corollary 15
on p. 110 of Hoyden [162] or Theorem 1.20 of Rudin [163].)Also note that, for Lipschitz functions on manifolds, integration by parts
holds (see Lemma 7.113 in Part I, for example).1.2. Convex functions on Euclidean spaces and their differen-
tiability.
It is standard in the convexity literature to consider functions which take
the value oo. In particular, given a locally convex set Cc IRn,I a function
f : C --+ IR U { oo} is said to be convex if for any line segment xy c C we
have for all s E (0, 1),
f (sx + (1 - s) y) ~sf (x) + (1 - s) f (y)
(with the operation + and relation ~ on IR U { oo} defined in the obvious
ways). The function f being convex is denoted by f E Conv ( C).
REMARK H.4. Given a convex set C C IRn and a convex function f :
C--+ IR, we may define J: IRn --+IR by
J (x) = { :(x)
if x EC,
if x E lRn - C.It is then easy to see that J E Conv (IRn).
If a convex function f is defined on a convex set in Euclidean space,
then it is a standard result that f is locally Lipschitz in the interior of the
set (see Theorem l(i) on p. 236 of [59] or Lemma 3.1.1 on p. 102 of [99]).
We shall give a proof of this result for f defined on Riemannian manifolds
(see Lemma H.21). Recall that the interior of a set ~ is denoted by int(~).PROPOSITION H.5 (Convex functions are locally Lipschitz). Any finite-
valued convex function f defined over a locally convex set C C IRn is locally
Lipschitz on int (C). Hence (by Rademacher's theorem) f is differentiable
almost everywhere on int ( C).
Concerning the convergence of convex functions, we have the following
general result (see Theorem 3.1.4 of [99]).
LEMMA H.6 (On convergence of sequences). If fk : IRn --+ IR U { oo} are
convex functions converging pointwise to a function f 00 : IRn --+ IR U { oo},
then f 00 is convex and fk converges uniformly to f oo on compact sets.Regarding the differentiability of functions, the following is equivalent
to Definition 7.115 in Part I.
(^1) Locally convex sets in Riemannian manifolds are defined in subsection 2.1.2 below.
'I