PropositionStatement: AI-adaptedProof: AI-generatedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-13

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A $C$ -Lipschitz map multiplies path length by at most $C$ ; isometries preserve length and scalar dilation multiplies it by the absolute scale

Statement

Let $γ : [a, b] \to R^{n}$ be a path and let $T : R^{n} \to R^{m}$ be Lipschitz with constant $K \geq 0$ . Then

$L (T \circ γ) \leq K L (γ)$

whenever $L (γ)$ is finite; if $L (γ) = + \infty$ the inequality is understood as the corresponding extended-real bound for $K > 0$ , while for $K = 0$ the composite is constant and has length zero.

If instead

$∥ T (x) - T (y) ∥_{2} = λ ∥ x - y ∥_{2}$

for every $x, y$ , then $L (T \circ γ) = λ L (γ)$ for $λ > 0$ , and it is zero for $λ = 0$ . In particular Euclidean isometries preserve length.

Facts & Assumptions

Given: The path and map in the statement.

[L1]

A Lipschitz map with constant $K$ satisfies $d (T x, T y) \leq K d (x, y)$ for every pair (Lipschitz map, $α$ -Hölder map for rational $0 < α \leq 1$ , and contraction).

[L2]

Isometries preserve every distance (Isometry, isometric embedding, and the subspace metric on a subset).

[L3]

Arc length is the supremum of sums of chord lengths (Paths in $R^{n}$ , inscribed polygonal sums, arc length as their supremum, and rectifiability).

Proof

technique · comparison

1.1

For every partition $P$ , applying [L1] to each chord and summing gives $ℓ_{P} (T \circ γ) \leq K ℓ_{P} (γ)$ .

givenL1L3

2.1

Taking suprema proves the Lipschitz estimate when $K > 0$ and also when $L (γ)$ is finite. If $K = 0$ , [L1] makes all images equal, so every polygonal sum is zero.

step 1.1L3

2.2

Under the similarity identity, every chordwise inequality in step 1.1 is an equality, so $ℓ_{P} (T \circ γ) = λ ℓ_{P} (γ)$ for every $P$ .

givenL3

3.1

Taking suprema gives exact scaling for $λ > 0$ ; for $λ = 0$ the map is constant on the trace and the length is zero. With $λ = 1$ , [L2] identifies the isometric case.

step 2.2L2L3∎

Depends on

Used by

The Koch curve is a uniform limit of polygonal paths of lengths (4/3)ⁿ but is not rectifiable Counterexample

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 102 results over 20 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click the chart to enlarge it.

Sources

U. Lang, Differential Geometry I, Section 1.1 (standard reference, not scraped)

A CC-Lipschitz map multiplies path length by at most CC; isometries preserve length and scalar dilation multiplies it by the absolute scale