- GRADIENTS OF CONVEX FUNCTIONS ON RIEMANNIAN MANIFOLDS 441
We remark that, in the proof of the Sharafutdinov retraction Theorem
H.59, we shall apply the above result a few times.3.3. Differentiability of convex functions on Riemannian man-
ifolds.
Let V be an open subset of (Mn, g). We say that a continuous functionf : V -+ JR has nonnegative Hessian in the support sense if for any
p E V and any c: > 0 there is a neighborhood U c V of p and a C^2 'local
lower barrier function' cp : U -+ JR such that cp (p) = f (p), f ( x) 2:: cp ( x) for
all x EU, and \l\lcp (p) 2:: -c:. We denote this by
Hess supp (f) 2:: 0.
Like convex functions on convex sets in Euclidean spaces, convex func-
tions on convex sets in Riemannian manifolds have the following differen-
tiability property. Note that Definition H. 7 regarding second derivatives
can easily be adapted to manifolds by using local coordinates. That is, a
function f on (Mn, g) is twice differentiable in the sense of Stolz at
x EM if for any C^00 local coordinates</> defined in a neighborhood of x we
have f o ¢-^1 is twice differentiable in the sense of Stolz at</> (x).
LEMMA H.39 (Convex functions have nonnegative Hessian). Let C c
M be a connected locally convex set with nonempty boundary and interior.
Suppose f : C -+ JR is convex. Then for any p E int ( C) we have
(H.29) Hess supp (f) 2:: 0
and hence f has second-order derivatives almost everywhere on int ( C).PROOF. For any p E int (C) and V E TpM with \VI = 1, by (H.21) we
have
f(expp(sV)) 2:f(p)+sDvf(p) fors;::::O.
An adaptation of the proof of Lemma H.37 yields
f ( expP ( s V)) 2:: f (p) - s (\! f (p) , V) ,where the generalized gradient ! f (p) is unique. Hence
cp ( expP ( s V)) ~ f (p) - (\! f (p) , s V) ·is a smooth function defined in a neighborhood of p and cp is a local lower
barrier function for the continuous function f at p. Since Hess ( cp) (p) = 0,
we conclude that Hess supp (f) (p) 2:: 0 and hence
Hess supp (J) 2:: 0 on int ( C).
Let { x} be normal coordinates centered at p. It follows from Lemma
7.122(iii) in Part I that there exists a C^2 function 'ljJ defined in a neighbor-
hood of p such that f + 'ljJ is a convex function of x, in the Euclidean sense,
for lxl sufficiently small. Hence, by the Aleksandrov theorem on convex
functions (see Lemma 7.117(ii) in Part I), the second derivative D^2 f exists
a.e. on int (C). D