456 H. CONVEX FUNCTIONS ON RIEMANNIAN MANIFOLDS
(ii) If f (x) < s::; f (y), then rs (y) = y, so that by Lemma H.52,
de (rs (x) ,rs (y)) =de (Ix (s) ,y)::; de (x,y).(iii) Ifs > f (y), then by Lemma H.49,
de (rs (x) 'rs (y)) =de bx (s) '/y (s)) ::; de (x, y).
This completes the proof of (H.49).
(2) This follows easily from the fact that rs (x) is uniformly continuous
in s and from the fact that for fixed s, rs ( x) is continuous in x. D
5. Notes and commentary
A classic reference for convex analysis is Rockafellar [160]. Another
reference is Hiriart-Urruty and Lemarechal [99].
§2. For Lemma H.15 see Lemma 1 in [171].§3. Given a locally Lipschitz function f : M ---+JR., we define the gen-
eralized Hessian to be the following difference quotient: for VE TxM,
(\7\7 1..f) (V, V) ~ limi
0nf f (iv (s)) + f (ti~ (-s)) -^2 f (x) E [-oo, +oo],
s-+ S
where /V: (-c:, c:)---+ M is the constant speed geodesic with iv (0) = V.
For example, consider the absolute value function f (x) = !xi on R Then
for 1 E TolR. we have at x = 0,
(\7\7 1..f) (1, 1) = liminf
2
1;1 = +oo.
s-+0 S
For Lemma H.37 see Lemma 3 in [171]. For Lemma H.38 see the corol-
lary to Lemma 3 of [171].
§4. We shall further discuss Sharafutdinov retraction in §1 of the next
appendix.
For Lemma H.41 see Lemma 4 in [171]. For Lemma H.45 see Lemma
5 in [171]. For Lemma H.48 see Lemma 2.1.3 in Petrunin [156] or Lemma
2.12 in Kapovitch, Petrunin, and Tuschmann [104]. For Lemma H.49 see
Lemma 6 in [171]. For Lemma H.52 see Lemma 7 in [171]. For Proposition
H.53 see Theorem 2 in [171]. For Lemma H.54 see Lemma 8 in [171]. For
Lemma H.55 see Lemma 9 in [171]. For Lemma H.57 see Lemma 10 in
[171]. For Lemma H.58 see Lemma 11 in [171]. For Theorem H.59 see
Theorem 3 in [1 71].