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Arc length is lower semicontinuous under uniform convergence of paths

Statement

Let $γ_{k}, γ : [a, b] \to R^{n}$ be paths, with $n \geq 1$ , and suppose

$\sup_{t \in [a, b]} ∥ γ_{k} (t) - γ (t) ∥_{2} ⟶ 0.$

Then

$L (γ) \leq {lim inf}_{k \to \infty} L (γ_{k})$

in the extended real line. In particular, a uniform limit can have smaller length than every approximating path, but not larger than their limiting lower length.

Facts & Assumptions

Given: The uniformly convergent sequence of paths.

[L1]

Uniform convergence uses one index for every point of the domain (Pointwise convergence, uniform convergence, and the uniformly Cauchy condition for sequences of real-valued functions); the displayed Euclidean-norm form gives convergence at each partition point.

[L2]

Euclidean norm and vector limits are compatible componentwise, so every fixed finite sum of chord norms converges term by term (A vector-valued function has a limit, or is continuous, if and only if each of its components does; with the algebra of continuous vector-valued functions).

[L3]

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

[L4]

The limit inferior is $\sup_{N} \inf_{k \geq N} x_{k}$ in the extended reals (Limit superior and limit inferior of a real sequence as $\inf_{n} \sup_{k \geq n} x_{k}$ and $\sup_{n} \inf_{k \geq n} x_{k}$ in $\overline{R}$ ).

Proof

technique · fixed-partition

1.1

Fix a partition $P = (t_{0}, \dots, t_{m})$ . By [L1], $γ_{k} (t_{i}) \to γ (t_{i})$ for each of its finitely many points.

givenL1

2.1

By [L2], every chord norm converges and hence $ℓ_{P} (γ_{k}) \to ℓ_{P} (γ)$ .

step 1.1L2

3.1

For every $k$ , [L3] gives $ℓ_{P} (γ_{k}) \leq L (γ_{k})$ . Passing to the limit inferior yields $ℓ_{P} (γ) \leq {lim inf}_{k} L (γ_{k})$ .

step 2.1L3L4

4.1

The right side is independent of $P$ . Taking the supremum of the left side over all partitions and using [L3] gives $L (γ) \leq {lim inf}_{k} L (γ_{k})$ .

step 3.1L3

5.1

The argument also covers an infinite right side or an infinite $L (γ)$ , because all suprema and the limit inferior are taken in the extended reals.

step 4.1L3L4∎

Depends on

Used by

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 120 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

J. Denzler, Calculus of Variations, Section 4.9 (standard reference, not scraped)