464 I. ASYMPTOTIC CONES AND SHARAFUTDINOV RETRACTIONPROOF. Define 1 = -bp - a2 : M --+ JR and
C = (-bp)-^1 ([a2, oo )) = 1-^1 ([O, oo )).
Since -bp is concave and proper, we have 1 E 12::stlo (C) and C is a compact
connected locally convex set with nonempty interior and boundary. ThenCs = {y E C : 1 (y) 2:: s}
= (-bp)-^1 ([s+a2,oo)).Now apply Theorem H.59 withs= al - a2 (so that Cs= (-bp)-^1 ([a1, oo)))
to obtain a d.istance-nonincreasing map= I'a1-a2l(-bp)-1(a 2 ): (-bp)-^1 (a2)--+ (-bp)-^1 (a1),
where I's: C--+ Cs is the distance-nonincreasing map defined by (H.48). DWe now define a global distance-nonincreasing retraction map. Fix any
integer mo satisfying mo :S supM (-bp) ~ (-bP)max· We define(I.5) Cmax (p) ~ {XE M: -bp (x) = (-bP)maJ,
Cm (p) ~ {x EM : -bp (x) 2:: m}for any integer m :S mo. Note that Cmax (p) C Cm (p) C Cm-1 (p) for any
m :S mo.
We may apply Theorem H.59 to the concave function 1 = -bp - mo,
C = Cm 0 (p)=1-^1 ([O,oo)), ands= (-bP)max -mo to obtain a distance-
nonincreasing retraction map ·
I' mo,max : Cm 0 (p) --+ Cmax (p).We may also apply Theorem H.59 to the concave function 1 = -bp - m,
C = Cm (p) = 1-^1 ([O, oo)), and s = 1 to obtain a distance-nonincreasing
retraction map
r m-1,m: Cm-1 (p)--+ Cm (p) for any m::; mo
(note that I' mo,maxlcmax(P) =id and I' m-1,mlcrn(P) =id).
Recall that Cmax (p) contains a soul of (Mn, g). Now we can define the
Sharafutdinov retraction map
Shar : M --+ Cmax (p).For any x E M, either x E Cm 0 (p) or there is a unique integer m ::; mo such
that x E Cm-1 (p) - Cm (p). We define
Shar (x) ~ {I' mo,max(x) ~f x E Cm 0 (p),
I' m,m+l o · · · o I' mo-1,mo o I' mo,max(x) if x E Cm-1(P) - Cm(p).
The following result is an easy consequence of the properties of r mo,max
and I'm-1,m·