1547845440-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_III__Chow_

410 G. ELEMENTARY ASPECTS OF METRIC GEOMETRY

function is not necessarily smooth. In fact, in this case, d ( ·, p) is a Lipschitz
function.^23
On Riemannian manifolds with sectional curvature bounded from below,
the Hessian of d ( ·, p) is bounded from above at smooth points. This suggests
that one study Lipschitz functions, in particular, functions with an upper
Hessian bound in the support sense, on Aleksandrov spaces. Although these
functions do not have derivatives everywhere, they behave nicely. When we
study such functions on an Aleksandrov space with curvature bounded from
below, another complication arises: the underlying topological space is not
a smooth manifold.
The following lemma generalizes the corresponding result in Riemannian
geometry regarding \Jd (the Gauss lemma).

LEMMA G.53 (Derivative of distance). Let X be an Aleksandrov space

of curvature ;:::: k and let p, q E X. Let a be a minimal geodesic from q to p

and let (3 be a unit speed minimal geodesic starting from q. Suppose that p

is not in the cut locus C (q). Then the following right derivative exists and

is given by

_i_/ d ( (3 (t)) = l' d (p, (3 (L\t)) - d (p, (3 (0)) = - / ( (3)

d

+ p,. lm A COS 4-q ct,.

t t=O .6.t-+O+ ut

For a proof of the above lemma, see Corollary 62 of [159] or Remark

4.5.12 in [18], which contain information about the case where q EC (q).

Now we give a definition of semi-concave functions, which correspond to
functions with a local upper Hessian bound.

DEFINITION G.54 (>.-concave function). Let X be an Aleksandrov space
of curvature ;:::: k without boundary, let n c X be an open subset, and let

. E JR. We say that a locally Lipschitz function f : n --+ IR is >.-concave if,
for any unit speed geodesic r : (a, b) --+ n, the function

is concave on (a, b).

1

s !-----+ f 0 I ( s) - ->.s^2

2

A function f : 0 --+ IR is called semi-concave if for every x E 0 there

exist a neighborhood U of x and ).. E IR such that flu is >.-concave. Note

that a C^2 function on a Riemannian manifold is semi-concave.

EXAMPLE G.55. Let Ji (x) = JxJ°', where a E (0, 2). Then, for any

. E IR, the functions r-+ Ji 01 (s)-~>.s^2 is not concave in any neighborhood

of 0. Thus, for a E [1, 2), the function Ji (x), which is Lipschitz for such a,

is not semi-concave.

We say th.at a locally Lipschitz function f : X --+ IR is >.-concave at a

point x EM if for every E > 0, the function

1 2

y !-----+ f (y) -

2

(>. + i::) d (y, x)

(^23) See subsection 1.1 for the definition of Lipschitz function.