418 H. CONVEX FUNCTIONS ON RIEMANNIAN MANIFOLDS
2.1.1. Interior tangent cone of a subset.
Let S be a subset in a Riemannian manifold (Mn,g) and let p ES. Wedefine the interior tangent cone by
(H · 4 ) i P S , -;--{v E 'I', P M ·. expP there (sV) exists E a int^6 (S) >^0 for such alls that E (0,6] } ·
REMARK H.11. The above definition of the interior tangent cone TpS is
one of a few, nearly equivalent, definitions of the tangent cone at p of the
set int (S) U {p }.
Of course, TpS may be empty even when int (S) is top-dimensional, i.e.,
even when int (S) # 0. For example, take(H.5) S ~ {(x,y): x 2: 0 and ./X :Sy :S 2./X} c IR^2
and take p = (0,0). Then TpS = 0 whereas TpS ={(0,y): y 2: O}, where
TpS denotes the tangent cone defined in (G.10).
We have the following properties of the interior tangent cone, which
follow directly from definition (H.4):
(1) If p E int (S), then
TpS =TpM.Note that if p ES\ int (S), then 0 ¢: TpS.
(2) The set TpS has a cone structure, i.e., for any VE TpS and r > 0
we haverV E TpS.
REMARK H.12. Assuming that the tangent cone TpS exists and assuming
that TpS naturally embeds in TpM (see Question G.21), we expect that
TpS CTpS·
When S is convex,^4 we have the following.LEMMA H.13 (Openness of TpS). If S is convex, then TpS is an open
subset of TpJV1..PROOF. (1) If p E int (S), then TpS = TpM and we are done.
(2) Suppose p E as. If v E TpS, then v -I 0 and there exists 6 E
(0, inj (p)) such thatexpP (6
1~
1) E int (S).Then there exists an open neighborhood U of 6Wr in TpM such that
expP (U) c int (S) and Uc B (o, inj (p)).
(^4) See subsection 2.1.2 for the definition of convex set.