1547845440-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_III__Chow_

430 H. CONVEX FUNCTIONS ON RIEMANNIAN MANIFOLDS

(ii) We compute for s > 0 and VE dom (Js) that when jsVI < c:,

IJs (V)I = f (expp (sV)) - f (p) :::;; Ld (expp (sV) ,p) :::;; LIVI.

s s

(iii) Since f is convex, for s > 0 small enough,

1 1

f (p) S 2 f ( expP ( s V)) + 2 f ( expP ( -s V))

and we have

f (expP (-sV)) - f (p) f (expP (sV)) - f (p)

- s < - s.

We conclude that

• (D-v f) (p) s (Dv f) (p)
  and (Dv f) (p) E (-oo, oo).
  Furthermore, we also have the following.

D

LEMMA H.26 (Convexity and upper semi-continuity of directional de-

rivative). Let f be a convex function on a connected locally convex set C in

(Mn,g).

(i) If (M,g) = IEn, then the difference quotient Js is convex in V.

Hence Dv f (p) is a convex function of V on TpC.

(ii) If f is Lipschitz in some neighborhood of p, then Dv f (p) is a con-

vex function of V on TpC.

(iii) The function

defined by

D f (p, V) ~ Dv f (p) ,

is upper semi-continuous.

PROOF. (i) For Vi, 112 E TpC and t E [O, 1], we compute on IEn that for

s > 0 small enough,

Js (tVi + (1 - t) 112)

f (p + s (tVi + (1-t) 112)) - f (p)

s

f (t (p + sVi) + (1 - t) (p + s112)) - f (p)

s

< tf (p + sVi) + (1 - t) f (p + s112) - f (p)

• s
  = tJS (Vi) + (1 - t) JS (112) ·

The convexity of Dv f (p) in V follows from taking the limit, as s --+ o+, of
the above inequality.

(ii) Let Is (u), u E [O, 1], be the minimal geodesic joining expP (sVi) and
expP (s112). Given t E (0, 1), let Is (us (t)) be the point on Is (u)i[o,l] which