1547845440-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_III__Chow_

434 H. CONVEX FUNCTIONS ON RIEMANNIAN MANIFOLDS

(iii) Let W be as in (H.16). If (Dminf) (p) :S 0, then

(H.19) (D\lf(p)f) (p) = - l\7f (P)l^2 ,

whereas if (Dminf) (p) 2: 0, then

(H.20) (D\lf(p)f) (p) = l\7f (p)l^2 ·

.(iv) If for some c > 0 and L < oo we have If (x) - f (p)I :S Ld (x,p)

for all x EB (p, c), then

l\7f (P)I :SL.

PROOF. (i) This follows directly from definition (H.17).

(ii) Consider the convex Lipschitz function f : JR.n --+ JR. defined by

f (x) = lxl at p = 0. Then W and \7 f (0) both may be any unit vector in

S 0 n-1.

(iii) We compute ·

(D\lf(p)f) (p) = (Dl(Dwf)(p)IW f) (p)

= l(Dwf) (p)/ (Dwf) (p)

=sign ((Dminf) (p)) · l\7f(p)1^2 ·

(iv) This follows from Lemma H.25(ii). D

If f is a convex function defined on a connected locally convex subset C

of M, then we say that p EC is a critical point off if (Dminf) (p) 2: 0.

LEMMA H.31 (Interior critical points of convex functions are local min-

ima). Let f: C--+ JR. be a convex function. Ifp E int (C) and (Dminf) (p) 2: 0,

then f attains a local minimum at p.

PROOF. Since f is convex, for V E s;-^1 and s 2: 0, we have that

f ( expP ( s V)) is a convex function of s. Hence for s E [O, s]

f ( expP ( s V)) 2: f (p) + s ( Dv f) (p)

provided expP (sV) EC. The lemma follows from

(Dv f) (p) 2: (Dminf) (p) 2: 0

for all VE s;-^1. D

More generally, for a convex function f : C --+ JR., if p E int (C), then

there is_ a neighborhood U of p in C n B (p, inj (p)) such that

(H.21)

for x EU.
Now we give a definition.