424 H. CONVEX FUNCTIONS ON RIEMANNIAN MANIFOLDS
Hence, for any t E [O, 1], the unit vector l=!~:~~:J is an interior point of
the convex set nm: ~ cm: n Brn;(O, 2)' where cm: c ]En is the minimal convex
set (cone) which contains (i.e., is the convex hull of) both
{ s Wm: : WJE E wr and s 2: 0} and { s WJE : WJE E W~ and s 2: 0}.
Note that under the pointed blow-up (i.e., as ).. ---+ o+), the geodesics
emanating from p of (U,p, )..-^2 g) approach (straight) rays emanating from
0 in ( TpM, 0, g (p)) in any Ck-norm and on any compact subset. For )..small enough, the image of nJE in (U,p, )..-^2 g) is very close to some minimal
convex subset of C n U which contains both
{ expP (sW): WE W1 ands E (0, 2.Al}
and{ expP (sW): WE W2 ands E (0, 2.Al}.
It £ o ll ows th a t 1(1-t)Vi (l-t)Vi +tVilg(p) +tVi lies in the interior of{WE s;-^1 : expP (sW) EC n U for alls E (0, 2.AJ}.
Hence there exists c1 > 0 such that expP (c1 ((1 - t) Vi+ tV2)) is an interior
point of C n U. From this we may easily derive (i).
(ii) As in (i), we replace co by a smaller positive number if necessary so
that there exists an open neighborhood U 2 c U of expP (co 112) such that
U2 c U2 c int ( C).
Let W2 be an open neighborhood of V2 in the unit sphere s;-^1 as defined
in (i). By the convexity of C, for any c E (O,c 0 ), the set{
0 < s < sw and expexp P (cVi) (sz)/ [O,sw] is a }
expexpp(10Vi) (sZ) : minimal geodesic joining expP (cVi) and a pointin the set { expP (rW): WE W2 and r E (0, co)}
is a cone. By the same blow-up argument as in (i), when c is small enough
(say c ::::;; c2 for some c2 E (0, co)), expP (c ((1-t) Vi+ tV2)) is an interior
point in the cone above. Hence
expP (s ((1-t) Vi+ tV2)) E int (C)
for 0 < S =S; c2. D
As a simple consequence of Lemma H.18, we have the following.
COROLLARY H.19 (Interior tangent cone of a locally convex set). Let
C CM be a locally convex set. Then TpC de.fined in (H.4) is a convex cone.
2.2. Convex functions on connected locally convex sets.
We recall some facts about convex functions on locally convex sets in
Riemannian manifolds and their directional derivatives.