PropositionStatement: AI-adaptedProof: AI-adaptedPipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-10

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Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

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Distance to a nonempty subset is one lipschitz

Statement

For nonempty $A$ in connected $M$ , $x \mapsto d_{g} (x, A)$ is finite and $1$ -Lipschitz. More generally this holds on a component $C$ with $A \cap C \neq \emptyset$ .

Facts & Assumptions

Given: $x, y \in C$ and $A \cap C \neq \emptyset$ .

[F1]

Distance from a point to a subset: For $A \subseteq M$ , define the distance to the subset by $d_{g} (x, A) = \inf_{a \in A} d_{g} (x, a)$ , with $\inf \emptyset = + \infty$ . Use def-extended-riemannian-distance-on-a-disconnected-manifold. If the component $C$ of $x$ meets $A$ , all cross-component terms are infinite and may be discarded, so $d_{g} (x, A) = \inf_{a \in A \cap C} d_{g} (x, a) < \infty$ . If $A \cap C = \emptyset$ , every term is infinite and the value is $+ \infty$ . In particular $d_{g} (x, A) = 0$ for $x \in A$ , and $d_{g} (x, {a}) = d_{g} (x, a)$ .

[F2]

Riemannian distance is a metric: $d_{g}$ is a finite metric on a connected Riemannian manifold.

Proof

technique · direct

1.1F1F2given

Fix one $a_{0} \in A \cap C$ . Both distances to the set are bounded above by the finite point distances to $a_{0}$ , and below by zero. For every $a \in A \cap C$ , the triangle inequality gives $d_{g} (x, a) \leq d_{g} (x, y) + d_{g} (y, a)$ . Taking infima yields $d_{g} (x, A) \leq d_{g} (x, y) + d_{g} (y, A)$ .

2.1F1F2step 1.1∎

Interchange $x, y$ and use symmetry to obtain the opposite inequality, so $∣ d_{g} (x, A) - d_{g} (y, A) ∣ \leq d_{g} (x, y)$ . All quantities subtracted are finite by step 1.1. On a component missing $A$ the extended value is $+ \infty$ throughout, with no real-valued Lipschitz assertion.

Source locator

Lee, Chapter 13, pp.337–340, Proposition 13.25, Lemma 13.28 and Theorem 13.29; finite piecewise $C^{1}$ refinements and pauses are treated explicitly here.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

8 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

John M. Lee, Introduction to Smooth Manifolds, second edition (standard reference, not scraped)