1547845440-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_III__Chow_

416 H. CONVEX FUNCTIONS ON RIEMANNIAN MANIFOLDS

DEFINITION H.7. Let U c ffi.n be an open set. A continuous function

f : U -t ffi. is twice differentiable in the sense of Stolz at x E U if there

exists an n-vector W and a symmetric n x n matrix M such that

f (x + y) = f (x) + W · Y + M (y, y) + o (1Yl^2 )

for x + y EU.

Recall from Lemma 7.117 in Part I (Aleksandrov's theorem) that a con-

vex function on a Euclidean ball is twice differentiable in the sense of Stolz

almost everywhere.

Although there are good reasons for allowing convex functions to take

the value oo, throughout most of this book, we shall assume that convex

functions are finite valued.

1.3. A property about convex sets in Euclidean space.

Let KC ffi.n be a closed convex set. Define the nearest point projec-

tion map

1fK: ffi.n -t K

by 1fK (x) being the unique closest point in K to x. Clearly 1fK o 1fK = 1fK

and 7fK (x) = x if and only if x E K. The following essentially says that a

closed convex set is on one side of its support planes.

LEMMA H.8. For any x E ffi.n and z EK we have

(x - 1fK (x), z - 1fK (x)) ::::; 0.

PROOF. Geometrically this is obvious, noting that the vector x-1fK (x)

is normal to a support plane. An analytic proof of the lemma may be found

for example in the proof of Theorem 3.1.1 on p. 47 in [99]. See also Lemma

10.36 in Part II. D

PROPOSITION H.9 (Monotonicity of the nearest point projection map).

Let K C ffi.n be a closed convex set. For any x, y E ffi.n we have

(1) (7rK is distance nonincreasing)

(H.l) 11fK (x) - 1fK (y)j :S Ix - YI,

(2) ( 1fK is monotone increasing)^2

(H.2) (7rK (x) - 1fK (y), X - y) 2:: 0.

PROOF. By Lemma H.8 we have for any x, y E ffi.n

(x - 1fK (x), 1fK (y) - 1fK (x)) :S 0,

(y - 1fK (y), 1fK (x) - 1fK (y)) :S 0,

so .that by summing we have

(y - x - 1fK (y) + 1fK (x) '1fK (x) - 1fK (y)) ::::; o.

(^2) Quoting p. 48 of [99].