To establish a firm footing for analysis, we find it
helpful assume a certain regularity to the manifold beyond mere
differentiability. For this purpose, we adopt the condition number
defined recently by Niyogi et al. [4].

[4]
Let MM be a compact submanifold of RNRN. The *condition number* of MM is defined as 1/τ1/τ,
where ττ is the largest number having the following
property: The open normal bundle about MM of radius rr
is imbedded in RNRN for all r<τr<τ.

The open normal bundle of radius rr at a point x∈Mx∈M
is simply the collection of all vectors of length <r<r anchored
at xx and with direction orthogonal to Tan x Tan x.

In addition to controlling local properties (such as curvature) of
the manifold, the condition number has a global effect as well,
ensuring that the manifold is self-avoiding. These notions are
made precise in several lemmata, which we repeat below for completeness.

[4]
If MM is a submanifold
of RNRN with condition number 1/τ1/τ, then the
norm of the second fundamental form is bounded by 1/τ1/τ
in all directions.

This implies that unit-speed geodesic paths on MM have
curvature bounded by 1/τ1/τ. The second lemma concerns the
twisting of tangent spaces.

[4]
Let MM be a submanifold
of RNRN with condition number 1/τ1/τ. Let p,q∈Mp,q∈M be two points with geodesic distance given by
dM(p,q)dM(p,q). Let θθ be the angle between the tangent
spaces Tan p Tan p and Tan q Tan q defined by cos(θ)=minu∈ Tan pmaxv∈ Tan q|〈u,v〉|cos(θ)=minu∈ Tan pmaxv∈ Tan q|〈u,v〉|. Then cos(θ)>1-1τdM(p,q)cos(θ)>1-1τdM(p,q).

The third lemma concerns self-avoidance of MM.

[4]
Let MM be a submanifold of
RNRN with condition number 1/τ1/τ. Let p,q∈Mp,q∈M be two points such that p-q2=dp-q2=d. Then for all
d≤τ/2d≤τ/2, the geodesic distance dM(p,q)dM(p,q) is
bounded by dM(p,q)≤τ-τ1-2d/τdM(p,q)≤τ-τ1-2d/τ.

From Lemma 3 we have an immediate corollary.

Let MM be a submanifold of RNRN with condition
number 1/τ1/τ. Let p,q∈Mp,q∈M be two points such
that p-q2=dp-q2=d. If d≤τ/2d≤τ/2, then d≥dM(p,q)-(dM(p,q))22τd≥dM(p,q)-(dM(p,q))22τ.

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