370 7. THE REDUCED DISTANCE(iii) For 0::::; .\::::; 1 and x,y EB (r - c), we compute
fi; (.Ax+ (1 - .A) y) = 2- r f (.A (x + z) + (1 - .A)(y + z)) 1] (~)dz
~}BW E:::;2- r (.\f(x+z)+(l-.\)f(y+z))17(~)dz
En jB(c) E
= .\fi; (x) + (1-.\) fi; (y) ·
(iv) See Theorem 6 on p. 630 of Evans [137]. D9.3. Functions with Hessian up:per bound. Recall that (.Mn, g)
is a complete Riemannian manifold. First we give a definition.DEFINITION 7.120 (Hessian upper bound in support sense). Let CE JR
and W c M be an open set. A continuous function f : W ----+.JR has the
Hessian upper bound C in the support sense if for any p E W and any
E > Q.there is a neighborhood U of panda 02 local upper barrier function
cp : U ----+ JR such that cp (p) = f (p), f (x) ::::; cp (x) for all x E U, and
"\l"\lcp (p)::::; C + E. We denote this by
Hess supp (!) s C.
REMARK 7.121. Clearly we can generalize the above definition to the
case where C : W ----+ JR is a function. However we shall only need the case
where C is a constant.
It is easy to see that when f is C^2 , Hess supp (!) s C implies '\JV f (p) s
C for each p E W. The following elementary lemma enables us to study the
differentiability properties of functions with a Hessian upper bound via the
theory of convex functions in Euclidean space. The idea is that we can add a
suitable smooth function to a function with a Hessian upper bound to make
it concave. Let Bp (r) c TpM denote the open ball of radius r centered at
0, and let B(p, r) C M denote the open geodesic ball of radius r centered
at p EM.
LEMMA 7.122 (Functions with a Hessian upper bound). Let WC M be
an open set, let f : W ----+ JR be a continuous function, and let p E W
(i) IfHesssupp (!)SC, then for any c > 0 there exists r > 0 and a C^2
function 'ljJ on B (p, r) such that Hess supp (f + 'ljJ) ::::; -c on B (p, r).
(ii) If Hess supp (!) s 0 on W, then -f is a convex function on (W, g) in
the Riemannian sense, i.e., the restriction of -f to each geodesic
segment in W is convex.
(iii) Let r S min { 1, injJp)} and x: B (p, r)----+ Bp (r) C TpM be normalcoordinates. If there exist constants C1, C2 < oo such that for each
q E B (p, r) there exists a local upper barrier function cpq for f at q
satisfying
JVcpqJ (q) s C1 and "\l'Vcpq (q) s C2g (q),