- CONNECTED LOCALLY CONVEX SUBSETS IN RIEMANNIAN MANIFOLDS 417
Hence, for any x, y E :!Rn,
(H.3) i11x (x) - 1fK (y)l^2 ::::; (7rK (x) - 1fK (y) 'x -y).
Part (2) immediately follows from this and also part (1) follows from this
and the Cauchy-Schwarz inequality:(7rK (x) - 1fK (y) ,x -y)::::; 17fK (x) - 1fK (y)l Ix -yl.
D
A simple consequence of (H.l) is the following Busemann-Feller the-orem (compare with Theorem H.59). Recall that the length of a path is
defined by (G.5).COROLLARY H.10 (Projection is length nonincreasing). Let Kc ]Rn be
a closed convex set and let ry be a rectifiable path in :!Rn. Then the projected
path 1fK o ry has length less than or equal to the length of ry.In the next section we develop tools in order to generalize this result to
convex sets in Riemannian manifolds.2. Connected locally convex subsets in Riemannian manifolds
In this section we study connected closed locally convex subsets in Rie-
mannian manifolds and the convex functions defined on them. These sets
are useful in the study of complete noncompact manifolds with nonnegative
sectional curvature.^3 Our main goal is to present some tools so that later
we may give a proof of Sharafutdinov's retraction theorem (see Theorem
H.59 below). As we shall see (Proposition H.17), connected closed locally
convex subsets in Riemannian manifolds are Aleksandrov spaces (see also
p. 821 of Plaut [159]). Hence the concepts and results given in this section
may help motivate the corresponding concepts and results for Aleksandrov
spaces, as we alluded to in the introduction to this appendix. One may view
the presentation below as an introduction to some of the techniques used in
studying Aleksandrov spaces, albeit in a much simpler form.
In this section (Mn, g) shall denote a connected complete Riemannian
manifold. Many of the results in this section are from Sharafutdinov [1 71].
Caveat: In §2-§4, we sometimes employ ad hoc techniques in the proofs
of the results contained therein; these proofs may not be the most efficient
proofs and these results are not the most general. The reader is encouraged
to dig into the literature to learn the related material.
2.1. Locally convex subsets in Riemannian manifolds.
In this subsection we discuss locally convex subsets in Riemannian man-
ifolds and their relation to Aleksandrov spaces.
(^3) For example, sublevel sets of Busemann functions in complete noncompact manifolds
with nonnegative sectional curvature satisfy a stronger condition than local convexity (see
Proposition I.15).