CorollaryStatement: Literature-sourcedProof: AI-adaptedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-13

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Every endpoint chord is no longer than the arc: $∥ γ (b) - γ (a) ∥_{2} \leq L (γ)$

Statement

Let $γ : [a, b] \to R^{n}$ be a path, with $n \geq 1$ . For every $a \leq u \leq v \leq b$ ,

$∥ γ (v) - γ (u) ∥_{2} \leq L_{[u, v]} (γ ∣_{[u, v]}) .$

This includes $u = v$ , when both sides are zero. In particular a path of length zero is constant.

Facts & Assumptions

Given: The path $γ$ and $u \leq v$ .

[L1]

For $u < v$ , the partition with point set ${u, v}$ has polygonal length $∥ γ (v) - γ (u) ∥_{2}$ , and arc length is the supremum of all polygonal lengths (Paths in $R^{n}$ , inscribed polygonal sums, arc length as their supremum, and rectifiability).

[L2]

On a singleton parameter interval, arc length is defined to be $0$ (Paths in $R^{n}$ , inscribed polygonal sums, arc length as their supremum, and rectifiability).

Proof

technique · direct

1.1

If $u < v$ , [L1] says the chord length is one member of the set whose supremum is the arc length, so it is at most that supremum.

givenL1

1.2

If $u = v$ , the chord is the zero vector and [L2] makes both sides zero.

givenL2

2.1

If the whole path has length zero, applying steps 1.1--1.2 to every $u \leq v$ makes every chord zero; separation for the Euclidean norm gives $γ (u) = γ (v)$ , so $γ$ is constant.

step 1.1step 1.2L1∎

Depends on

Paths in $\mathbb{R}^n$, inscribed polygonal sums, arc length as their supremum, and rectifiability

Used by

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 56 results over 13 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

T. M. Apostol, Mathematical Analysis, Section 6.10 (standard reference, not scraped)

Every endpoint chord is no longer than the arc: ∥γ(b)−γ(a)∥2≤L(γ)∥γ(b)−γ(a)∥2​≤L(γ)