Tailored-LEN: Canonical Path Parametrization

• Tailored-LEN is a functional that defines an alternative length for continuous paths by integrating stripwise projected diameters, offering a canonical parameterization.
• The framework guarantees key properties such as invariance under Euclidean isometries, subadditivity on subpaths, and minimality for straight segments.
• Tailored-LEN enables a standard reparametrization for path classes, facilitating criteria for equicontinuity and compactness in families where traditional rectifiability fails.

Tailored-LEN is the $\len$-based framework developed in the work of Hoehn, Oversteegen, and Tymchatyn for canonically parametrizing continuous paths in $\mathbb{R}^n$ by an alternative notion of length that is defined for every continuous path, is invariant under Euclidean isometries, is monotone on subpaths, and is continuous in the uniform topology (Hoehn et al., 2013). In the formulation summarized for $n=2$, the construction replaces Euclidean arc length by an integral of weighted stripwise projected diameters; this yields a standard parametrization for every non-constant path, a homeomorphic model for path classes modulo orientation-preserving retracing, criteria for equicontinuity and compactness, and a canonical extension procedure for certain laminations of arcs.

1. Definition of the $\len$ functional

For the planar case, the construction begins with the horizontal unit-width strips

$$S_j=\{\,a+ib:\;a\in\mathbb{R},\;b\in[j,j+1]\,\}, \qquad j\in\mathbb{Z}.$$

Given parameters

$x\in[0,1],\quad t\in[0,1],\quad \mu\in(0,1],$

one rotates by angle $\pi t$, scales by $\mu$, and translates vertically by $x$, obtaining

$$S^{x,t,\mu}_j=\mu\,e^{\,i\pi t}\!\bigl(S_j+i\,x\bigr).$$

For each triple $\mathbb{R}^n$0, the plane is therefore foliated by the disjoint strips $\mathbb{R}^n$1 (Hoehn et al., 2013).

If $\mathbb{R}^n$2 is continuous, one considers, for each $\mathbb{R}^n$3, the connected components $\mathbb{R}^n$4 of the preimage $\mathbb{R}^n$5. Each such component has a diameter under orthogonal projection to the line of direction $\mathbb{R}^n$6, denoted

$\mathbb{R}^n$7

The components with nonzero projected diameter are enumerated in nonincreasing order as $\mathbb{R}^n$8, and the associated strip-sum is defined by

$\mathbb{R}^n$9

The functional $n=2$0 is then

$n=2$1

A semicontinuity argument shows that $n=2$2 is Lebesgue-integrable and that $n=2$3 for every non-constant $n=2$4. The $n=2$5-dimensional case is stated to be analogous, replacing strips by parallel $n=2$6-planes. This suggests that Tailored-LEN should be understood not as a Euclidean metric replacement in the usual differential-geometric sense, but as a geometric averaging procedure over foliations and projected diameters.

2. Structural properties of $n=2$7

The functional satisfies several basic properties. If $n=2$8 is a Euclidean isometry, then

$n=2$9

If $\len$0 is defined on a dendrite $\len$1 and $\len$2 is a closed subcontinuum, then

$\len$3

with equality if and only if $\len$4 is constant on each component of $\len$5. If $\len$6 and $\len$7, then

$\len$8

These assertions encode invariance, monotonicity on subpaths, and subadditivity (Hoehn et al., 2013).

The minimality property is formulated for straight segments. If $\len$9 and $$S_j=\{\,a+ib:\;a\in\mathbb{R},\;b\in[j,j+1]\,\}, \qquad j\in\mathbb{Z}.$$0 is any path joining them, then

$$S_j=\{\,a+ib:\;a\in\mathbb{R},\;b\in[j,j+1]\,\}, \qquad j\in\mathbb{Z}.$$1

with strict inequality unless $$S_j=\{\,a+ib:\;a\in\mathbb{R},\;b\in[j,j+1]\,\}, \qquad j\in\mathbb{Z}.$$2 is exactly the monotone parametrization of that line segment. This places straight segments in the role of $$S_j=\{\,a+ib:\;a\in\mathbb{R},\;b\in[j,j+1]\,\}, \qquad j\in\mathbb{Z}.$$3-geodesics among all paths with fixed endpoints.

Every continuous path $$S_j=\{\,a+ib:\;a\in\mathbb{R},\;b\in[j,j+1]\,\}, \qquad j\in\mathbb{Z}.$$4 satisfies

$$S_j=\{\,a+ib:\;a\in\mathbb{R},\;b\in[j,j+1]\,\}, \qquad j\in\mathbb{Z}.$$5

and

$$S_j=\{\,a+ib:\;a\in\mathbb{R},\;b\in[j,j+1]\,\}, \qquad j\in\mathbb{Z}.$$6

is continuous in the uniform-norm topology. The combination of finiteness for all continuous paths and continuity in the uniform metric is a defining distinction from Euclidean path length. A plausible implication is that $$S_j=\{\,a+ib:\;a\in\mathbb{R},\;b\in[j,j+1]\,\}, \qquad j\in\mathbb{Z}.$$7 is particularly well suited to topological families of paths where rectifiability is unavailable or unstable.

3. Canonical parametrization and path classes

Two paths $$S_j=\{\,a+ib:\;a\in\mathbb{R},\;b\in[j,j+1]\,\}, \qquad j\in\mathbb{Z}.$$8 are declared equivalent, written $$S_j=\{\,a+ib:\;a\in\mathbb{R},\;b\in[j,j+1]\,\}, \qquad j\in\mathbb{Z}.$$9, when they trace the same image with the same orientation and constant pieces are collapsed. For a non-constant path $x\in[0,1],\quad t\in[0,1],\quad \mu\in(0,1],$0, the map

$x\in[0,1],\quad t\in[0,1],\quad \mu\in(0,1],$1

is continuous and non-decreasing into $x\in[0,1],\quad t\in[0,1],\quad \mu\in(0,1],$2. After rescaling the target back to $x\in[0,1],\quad t\in[0,1],\quad \mu\in(0,1],$3, one obtains the unique reparametrization

$x\in[0,1],\quad t\in[0,1],\quad \mu\in(0,1],$4

This is the standard, or $x\in[0,1],\quad t\in[0,1],\quad \mu\in(0,1],$5, parametrization (Hoehn et al., 2013).

The standard parametrization satisfies $x\in[0,1],\quad t\in[0,1],\quad \mu\in(0,1],$6 and is uniquely determined by the equivalence class $x\in[0,1],\quad t\in[0,1],\quad \mu\in(0,1],$7. If $x\in[0,1],\quad t\in[0,1],\quad \mu\in(0,1],$8 denotes the set of all equivalence classes, then the map

$x\in[0,1],\quad t\in[0,1],\quad \mu\in(0,1],$9

induces a homeomorphism between $\pi t$0, equipped with its natural quotient topology, and the closed subspace

$\pi t$1

consisting exactly of those paths for which

$\pi t$2

This is the central canonicalization theorem of the framework. Rather than selecting a representative by speed normalization in the Euclidean sense, Tailored-LEN selects a representative by linearizing accumulated $\pi t$3. The result is a topological normal form for oriented path classes.

4. Equicontinuity and compactness in the $\pi t$4-framework

Let $\pi t$5 be a family of path classes, and let

$\pi t$6

be the corresponding family of standard parametrizations. The equicontinuity criterion states that $\pi t$7 is equicontinuous in $\pi t$8 if and only if the following condition holds: for each $\pi t$9 there exists $\mu$0 such that no member of $\mu$1 admits more than $\mu$2 disjoint subintervals whose images have diameter at least $\mu$3 (Hoehn et al., 2013).

The compactness criterion strengthens this. A set $\mu$4 has compact closure if and only if two conditions are satisfied: the set of initial points

$\mu$5

is bounded, and $\mu$6 satisfies the same $\mu$7-long-pieces condition. In particular, after passage to standard parametrizations, one recovers Ascoli–Arzelà in the $\mu$8-framework.

These criteria are notable because they avoid direct reliance on Euclidean arc-length bounds. The controlling quantity is combinatorial-geometric: the number of disjoint subintervals carrying image diameter above a fixed threshold. This suggests that Tailored-LEN provides a compactness theory adapted to non-rectifiable path families, where classical bounded-variation hypotheses would be too restrictive.

5. Estimates and representative examples

Several explicit estimates and examples clarify the scale of $\mu$9. Any straight-line segment $x$0 has the least possible $x$1 among all paths joining $x$2 to $x$3. For the unit segment $x$4 in the $x$5-direction, one computes directly that

$x$6

At the same time, every path still satisfies $x$7 (Hoehn et al., 2013).

For the winding example

$x$8

which traverses the unit circle $x$9 times, one has

$$S^{x,t,\mu}_j=\mu\,e^{\,i\pi t}\!\bigl(S_j+i\,x\bigr).$$0

while each individual $$S^{x,t,\mu}_j=\mu\,e^{\,i\pi t}\!\bigl(S_j+i\,x\bigr).$$1 remains strictly less than $$S^{x,t,\mu}_j=\mu\,e^{\,i\pi t}\!\bigl(S_j+i\,x\bigr).$$2. This indicates that repeated geometric complexity can asymptotically saturate the normalization range without ever exceeding it.

There is also a diameter comparison for small images. Whenever

$$S^{x,t,\mu}_j=\mu\,e^{\,i\pi t}\!\bigl(S_j+i\,x\bigr).$$3

there exist constants $$S^{x,t,\mu}_j=\mu\,e^{\,i\pi t}\!\bigl(S_j+i\,x\bigr).$$4, depending only on $$S^{x,t,\mu}_j=\mu\,e^{\,i\pi t}\!\bigl(S_j+i\,x\bigr).$$5, such that

$$S^{x,t,\mu}_j=\mu\,e^{\,i\pi t}\!\bigl(S_j+i\,x\bigr).$$6

This yields a local comparability between $$S^{x,t,\mu}_j=\mu\,e^{\,i\pi t}\!\bigl(S_j+i\,x\bigr).$$7 and image diameter. A plausible implication is that, at sufficiently small spatial scale, Tailored-LEN behaves like a normalized geometric size functional, while at larger scale it retains sensitivity to repeated traversals and path structure.

6. Midpoint–$$S^{x,t,\mu}_j=\mu\,e^{\,i\pi t}\!\bigl(S_j+i\,x\bigr).$$8 parametrization and lamination homeomorphisms

Section 4 of the summarized work introduces a second parametrization, the midpoint–$$S^{x,t,\mu}_j=\mu\,e^{\,i\pi t}\!\bigl(S_j+i\,x\bigr).$$9 parametrization $\mathbb{R}^n$00, which has the additional property that it commutes with reversal of orientation (Hoehn et al., 2013). This refinement is used for laminations.

A lamination $\mathbb{R}^n$01 of a planar set $\mathbb{R}^n$02 is described as a family of arcs filling $\mathbb{R}^n$03 so that they meet only in endpoints, together with suitable local-finiteness and compactness axioms. If $\mathbb{R}^n$04 is a bijection from the endpoint set $\mathbb{R}^n$05 to the endpoint set of another lamination $\mathbb{R}^n$06, then Theorem 4.2 extends $\mathbb{R}^n$07 uniquely to a continuous map

$\mathbb{R}^n$08

which, on each arc $\mathbb{R}^n$09, is given in coordinates by

$\mathbb{R}^n$10

where $\mathbb{R}^n$11 and $\mathbb{R}^n$12 are chosen so that $\mathbb{R}^n$13, $\mathbb{R}^n$14, and $\mathbb{R}^n$15.

This extension is a homeomorphism provided $\mathbb{R}^n$16 is one, and Theorem 4.3 states that the resulting lamination-homeomorphism depends continuously on the data $\mathbb{R}^n$17. In this part of the theory, Tailored-LEN serves not merely as a parametrization device for individual paths but as a coordinate mechanism for transferring entire arc-families through endpoint data. The significance is that canonical path coordinates become sufficient to construct global homeomorphisms in a structured topological setting.