- ALEKSANDROV SPACES WITH CURVATURE BOUNDED FROM BELOW 403
(2) Hinge version. Let L be a hinge^16 in M with vertices (p, q, r), sides
qr and rp which are geodesics, and interior angle Lqrp E [O, 7r]. Suppose that
qr is minimal and that L ( rp) ::::; 7r / Jk if k > 0.
If L' is a geodesic hinge with vertices (p', q^1 , r') in the complete simply-
connected surface of constant Gauss curvature k with the same side lengths
and same angle, then
d (p, q) ::::; d (p', q').
(3) Opposite side version. For any geodesic triangle 6.pqr in M and
points Erp, we have
d(q,s) 2: d(q,s),
where 6.pqf is the k-comparison triangle of 6.pqr and s E f p is the point
with d (p, s) = d (p, s).REMARK G.34. There are other versions of the Toponogov comparison
theorem. For example, there is the Toponogov monotonicity principle; see
Lemma G.38 below.
2.2. Aleksandrov spaces with curvature bounded from below.
With the above motivations, we make the following definitions.
A (geodesic) triangle in a complete length space (X, £) consists of
three points p, q, r E X and three shortest paths qr, rp, pq joining these
points. The points p, q, r are called the vertices of the triangle and the
paths qr, rp, pq are called the sides of the triangle. We denote this triangle
by 6.pqr. Since (X, £)is complete, given any three points in X, there exists
a triangle with these three points as its vertices. A pair of geodesic segments
with a common vertex is called a hinge.
Given a triangle 6.pqr in a complete length space (X, £), with
£, (qr) + £ ( rp) + £ (pq) < 27r / ../k
if k > 0, there exists a triangle Dpfjr in the complete simply-connected
surface M~ of constant Gauss curvature k with the same side lengths. This
triangle is unique up to isometry. As in the Riemannian case, we call the
triangle 6.pfjf the k-comparison triangle of 6.pqr.
There are several equivalent definitions of Aleksandrov space with cur-
vature bounded from below; we give the following. The advantage of this
definition is that it only uses length.
DEFINITION G.35 (Aleksandrov space-local version). Given k E JR,
a complete locally compact^17 length space (X, £) (with strictly intrinsic
metric d.c) is an Aleksandrov space of curvature 2: kif for every x EX
(^16) See subsection 2.2 below for the definition of hinge in an Aleksandrov space.
(^17) Local compactness is not assumed in [18] and [159], whereas it is assumed in [19],
[175], and [156].