- CONNECTED LOCALLY CONVEX SUBSETS IN RIEMANNIAN MANIFOLDS 425
2.2.1. Definition of convex functions.Let C C M be a connected locally convex set. We say that a function
f : C ---+ IR is convex if for any x, y E C and any constant speed geodesic
"( : [a, b] ---+ C (the geodesic is contained in C) joining x to y, the composite
function f o "( : [a, b] ---+ IR is convex, that is, for any u1, u2 E [a, b] and
t E [O, 1] we have^6
f O "( ( ( 1 - t) U 1 + tu2) '.S ( 1 - t) f O "( ( U 1) + t f O "( ( Uz).
We say that f : C ---+ IR is concave if -f is convex.
The above definition of the convexity (concavity) of a function makes
sense in an Aleksandrov space without boundary (in the definition, one
simply replaces 'constant speed geodesic' by 'unit speed shortest path').^7
Let ¢:I---+ IR, where IC IR is an interval, be a convex function. For all
so, s1 EI with s1 > so, the right derivative d~~ (so) exists and(H. 7 ) de/> (so) :S </> (s1) - ¢(so)
ds+ s1 - so
since the RHS is a monotonically nondecreasing function of s1 for fixed so.
Moreover, the function s H i!F ( s) is also monotonically nondecreasing.
Conversely to the first statement above, if a Lipschitz function ¢ : I ---+ IRsatisfies (H.7) for all so, s1 E I with s1 > so in I, then¢ is concave.
REMARK H.20. Let f : C ---+ IR be a convex function. Note that if
"( : I ---+ C is a unit speed geodesic, then
d
s 1-----t ds+ (J o "() ( s)
is a monotonically nondecreasing function.
2.2.2. Convex functions are locally Lipschitz.
We prove that convex functions .on connected locally convex sets in Rie-
mannian manifolds are locally Lipschitz in the interior of the set, which is
the analogue of Proposition H.5 ..LEMMA H.21 (Convex functions are locally Lipschitz). Let C C (Mn, g)
be a connected locally convex set. If f : C ---+ IR is a convex function, then f
is locally Lipschitz in int ( C).
PROOF. We divide the proof into three steps.
Step 1. f is locally bounded from above in int ( C). For any p E int ( C),
let
r E (O,inj (p) /4)
be such that
(1) B (p, 2r) is both convex and contained in C,
(2) inj (q) > 2r for q EB (p, r), and(^6) This definition is opposite to the definition of convexity used by Sharafutdinov.
(^7) In the literature, >..-concave with >.. = 0 is the same as concave.