1547845440-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_III__Chow_

414 H. CONVEX FUNCTIONS ON RIEMANNIAN MANIFOLDS

for all x 1 , x 2 E X. The Lipschitz constant (or dilatation) off is the

infimum of all such C:

Lip (J) ~inf { C: Vx1, x2 EX, dy (J (x1), f (x2)) :::; Cdx (x1, x2)}.

A map f : X -+ Y is locally Lipschitz if for every x E X there exists a

neighborhood U of x such that flu is Lipschitz. The dilatation off at x

is defined to be

dil x (J) ~ inf {Lip ( f I u) : U is a neighborhood of x}.

PROBLEM H.l. Under what conditions do we have the equality

Lip (f) = sup dil x (J)?

xEX

Exercise 1.4.5 of Burago; Burago, and Ivanov [18] says that this equality

is true if X = JR or X = 51 , where 51 is endowed with the intrinsic (arc

length) metric.

A function f on a differentiable manifold Mn is called locally Lipschitz

if it is locaHy Lipschitz with respect to some Riemannian metric g on M.

Note that the property of a function being locally Lipschitz is independent

of the metric. Hence, locally, a locally Lipschitz function on a manifold is

in essence a Lipschitz function on some Euclidean ball.

Recall that given a Riemannian manifold (Mn, g), there exists an associ-

ated Riemannian measureμ. A set Sc M has measure zero ifμ (S) = 0.

The property of a set having measure zero is independent of the metric g.

We say that a property on a differentiable manifold holds almost every-

where ( a.e.) if the property holds on the complement of a set with measure

zero with respect to some Riemannian measure. Locally Lipschitz functions

on differentiable manifolds have the following property (see Theorem 2 in

§3.1.2 on p. 81 of Evans and Gariepy [59] or Theorem 3 on p. 250 of Stein

[176]).

PROPOSITION H.2 (Rademacher). If Mn is a differentiable manifold and

if f : M -+ JR is a locally Lipschitz function, then f is differentiable almost

everywhere.

In particular, the distance function of a Riemannian manifold, which
has Lipschitz constant 1, is differentiable a.e. (alternatively, the distance
function to a point p is 000 outside the cut locus of p, which has measure
zero).
· When the manifold M is 1-dimensional, a Lipschitz function can be
expressed as the integral of its derivative.

LEMMA H.3 (Fundamental theorem of calculus). If f is a Lipschitz func-

tion defined on an interval [a, b], then for any x E [a, b],

f (x) = f (a)+ 1x f' (s) ds,