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A path in $R^{n}$ is rectifiable exactly when every coordinate has bounded variation

Statement

Let $n \geq 1$ , let $a \leq b$ , and let $γ : [a, b] \to R^{n}$ be a path, with coordinate functions $γ_{j} (t) : = γ (t) (j)$ for $j < n$ , so that $γ = (γ_{0}, \dots, γ_{n - 1})$ (The Euclidean inner product $⟨ x, y ⟩ = \sum_{k < n} x_{k} y_{k}$ on $R^{n}$ ). Then $γ$ is rectifiable if and only if every coordinate function $γ_{j}$ has bounded variation. When these conditions hold,

$\max_{j < n} {Var}_{[a, b]} (γ_{j}) \leq L_{[a, b]} (γ) \leq \sum_{j < n} {Var}_{[a, b]} (γ_{j}) .$

For $a = b$ , every term in this display is zero.

Facts & Assumptions

Given: The path $γ$ .

[L1]

The standard unit vectors are indexed by $j < n$ and satisfy $z = \sum_{j < n} z_{j} e_{j}$ , $⟨ z, e_{j} ⟩ = z_{j}$ , and $∥ e_{j} ∥_{2} = 1$ ; the Euclidean norm satisfies Cauchy--Schwarz, homogeneity, and the triangle inequality (The Euclidean inner product $⟨ x, y ⟩ = \sum_{k < n} x_{k} y_{k}$ on $R^{n}$ , The standard list $e : n \to F^{n}$ with $e_{i} (i) = 1_{F}$ and $e_{i} (j) = 0_{F}$ for $j \neq i$ is an ordered basis of $F^{n}$ ; hence $\dim_{F} F^{n} = n$ , and $F^{0}$ is the zero space with basis $\emptyset$ and dimension $0$ , Cauchy-Schwarz $∣ ⟨ x, y ⟩ ∣ \leq ∥ x ∥_{2} ∥ y ∥_{2}$ with its equality case, the triangle inequality for $∥ \cdot ∥_{2}$ , the parallelogram law and polarisation).

[L2]

Total variation is the supremum over partition sums $\sum ∣ f (t_{i + 1}) - f (t_{i}) ∣$ , with singleton variation defined as zero (Bounded variation and total variation on an interval).

[L3]

Arc length is the supremum over the corresponding sums of Euclidean chord lengths; rectifiability means that set is bounded (Paths in $R^{n}$ , inscribed polygonal sums, arc length as their supremum, and rectifiability).

Proof

technique · comparison

1.1givenL1L2L3

Cauchy--Schwarz in [L1] gives $∣ z_{j} ∣ = ∣ ⟨ z, e_{j} ⟩ ∣ \leq ∥ z ∥_{2}$ . Applying this to every chord of every partition gives $V (γ_{j}, P) \leq ℓ_{P} (γ)$ .

1.2givenL1L2L3

Conversely, $z = \sum_{j < n} z_{j} e_{j}$ and the norm axioms in [L1] give $∥ z ∥_{2} \leq \sum_{j < n} ∣ z_{j} ∣$ . Applying this to every chord gives $ℓ_{P} (γ) \leq \sum_{j < n} V (γ_{j}, P) \leq \sum_{j < n} Var (γ_{j})$ for every partition.

2.1step 1.1L2L3

If $γ$ is rectifiable, taking suprema in step 1.1 gives $Var (γ_{j}) \leq L (γ)$ for every $j$ , so all coordinates have bounded variation and the left displayed bound holds.

2.2step 1.2L3

If every coordinate has bounded variation, the final real number in step 1.2 bounds all polygonal sums. Hence $γ$ is rectifiable, and taking the supremum gives the right displayed bound.

3.1L2L3∎

If $a = b$ , the singleton conventions in [L2] and [L3] make all quantities zero, so both directions and both bounds remain valid.

Depends on

Dependency tree · 2 levels

Sources

A. R. Shastri, Metric Spaces, Section 5

A path in RnRn is rectifiable exactly when every coordinate has bounded variation