398 G. ELEMENTARY ASPECTS OF METRIC GEOMETRY(TpX, dTpX) exist. Does there exist a (natural) embedding i : TpS ---+ TpX
such that l * ( dTpX) = dTpS?
QUESTION G.22 (Existence of tangent cone of locally convex subset).
Let (Mn, g) be a Riemannian manifold. If S is a locally convex subset of M(see subsection 2.1.2 in Appendix H below for the definition) and if p ES,
then does the tangent cone TpS exist?
EXERCISE G.23 (Invariant of the tangent cone under rescalings). Showthat for any c > 0, (TpX, cdp, Op) is isometric to (TpX, dp, Op)·
EXERCISE G.24 (Nonuniqueness of tangent-type cone limits). Give an
example of a connected compact metric space (X, d) and a point p EX such
that there exist sequences ai ---+ oo and /3i ---+ oo such that the Gromov-
Hausdorff limitsboth exist but are not isometric.
1.3.2. Asymptotic cone of a boundedly compact metric space.
The Gromov-Hausdorff asymptotic cone (or tangent cone at
oo) of a boundedly compact metric space (X, d) is the pointed Gromov-
Hausdorff limit
(G.11)provided this limit exists and is independent (up to isometry) of the sequence
{wi}· Justifying the notation, the Gromov-Hausdorff asymptotic cone is
independent of the choice of p (see Proposition 8.2.8 of [18]).^12 Therefore,
by its definition, the asymptotic cone is unique when it exists. Note that
the asymptotic cone reflects the geometry at infinity of X.
EXERCISE G.25 (Invariance of the asymptotic cone under rescalings).Show that for any c > 0, (AX, cdAx, 0) is isometric to (AX, dAx, 0).
EXAMPLE G.26 (Examples of asymptotic cones which are half-lines).
(1) The asymptotic cone of a curvature pinching set in a vector space
of algebraic curvature operators, as in §7 of Chapter 11 in Part II,
is a ray.
(2) The asymptotic cone of a complete noncompact Riemannian man-
ifold with aperture equal to zero is a half-line.^13
EXAMPLE G.27. Note that, for hyperbolic space IfP, n 2: 2, the Gromov-
Hausdorff asymptotic cone does not exist; intuitively, the reason for this is
that the hyperbolic metric at a point 'grows too fast' in terms of the dis-
tance of the point to the origin (see Exercise 8.2.13 and the paragraph after(^12) That is, if the limit in (G.11) exists for some p E X, then the limit exists for all
p E X and it is independent of the choice of p.
(^13) See §18 of [92] for the definition of aperture.