- APpROXIMATE BUSEMANN-FELLER THEOREM 485
is a diffeomorphism, where
Ns,.-1-l ~{av (x): x E 1-l and expx (sv (x)) c M+ for 0 :S s :Sa< c:}.
Moreover, for every x E N;,tA,i-;,) (1-l), there exists a unique point Jr (x) E 1-l
closest to x. This point Jr ( x) also has the property that the unit speed
geodesic a7r(x) I [O,d(x,1-l)] is minimal and joins 1r (x) to x. It is easy to see that
n is smooth. DREMARK I.43. Note that we have shown that 1-lr is a smooth hypersur-face for 0 :Sr< i (A, K,).
The following gives an almost distance-decreasing property of the nearest
point projection map in a small tubular neighborhood. It may be thought of
as an approximate version of the Busemann-Feller theorem (see Corollary
H.10) for closed convex sets. We would like to thank Deane Yang for showing
us both the statement and the proof of the theorem.
THEOREM I.44 (Approximate Busemann-Feller theorem). Let (Mn,g)and 1-ln-l c M be as above, so that in particular we have sect (g) :S K,^2 and
II1-l :'.'.': -AI1-l· For any c: E (0, i (A, K,)], where i (A, K,) > 0 is given by (I.46),
we have for any smooth path / C N/ (1-l) that
(
L (no 1) :S cos (c:K,) - A K, sin (c:K,) )-1 L (1).
PROOF. Let 'Y : [O, 1] --+ N;,tA,i-;,) (1-l) be ~smooth path and let no 'Y be
its projection into 1-l. Without loss of generality, by reparametrizing /, we
may assume that Jr o 'Y : [ 0, 1] --+ 1-l has constant _speed ( = L (Jr o /)). Define
a smooth map (homotopy) ·
r: [o, 1] x [o, 1] --+ NitA,i-;,) (1-l)
by
r (s, t)-;--exp7r 0 'Y(t) sd (no 'Y (t), 'Y (t)) v (no/ (t)).Then we have
r(O,·) =1rO/,
r(l,·)=1,
and, for each t, the paths f-t r (s, t) is a constant speed geodesic. Let
ar arrs~ as and rt~ at.
Note that
(I.47)
for all t since rt (0, t) = It (n 01) (t) is tangent to 1-l whereas(I.48) rs ( o, t) = d ( n o 1 ( t) , 1 ( t)) v ( n o 1 ( t))
is normal to 1-l.