ExampleConstruction: Literature-sourcedVerification: AI-adaptedPipeline-generated

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The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

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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Separated and nonseparated initial supports

Example

Assume AC. In $Ω = R^{2}$ , let $σ$ be normalized arc-length probability on the unit circle $S^{1} = {x : ∣ x ∣ = 1}$ and let $δ_{0}$ be the point mass at the origin. For radii $r_{a} = 1 / 2$ and $r_{b} = 2$ , the support of $σ$ is contained in the ball $B (0, r_{b})$ and separates the two spheres $r_{a} S^{1}$ and $r_{b} S^{1}$ : every continuous path from one sphere to the other meets $S^{1}$ . In contrast, the support of $δ_{0}$ is contained in $B (0, r_{b})$ but does not separate those spheres. These are only checks of the topological initialization condition; neither example asserts that any flow converges or that its limit is globally minimizing.

Example references

Wasserstein gradient flow for a feature-measure energy

Feature-measure energy and its regularity assumptions

Support of a finite Borel measure on the plane

The Axiom of Choice

AC implies DC implies countable choice

Chizat and Bach, Theorem 3.3

Ambrosio, Gigli, and Savaré, Chapter 11

Facts & Assumptions

Given: AC and the two probability measures in the Example.

[F1]

The support of a finite Borel measure is the closed set of points whose every open neighborhood has positive measure. Support of a finite Borel measure on the plane

[F2]

A continuous real-valued function on an interval takes every value between its endpoint values. Intermediate value theorem, by bisection with a canonical left-half rule: a continuous function on $[a, b]$ takes every value between $f (a)$ and $f (b)$

Proof

Given: The measures and radii in the Example.

1.1F1given

Every open ball centered at a point of $S^{1}$ intersects an arc of positive arc length, so [F1] gives $supp σ = S^{1}$ . Since every point of $S^{1}$ has norm $1 < 2$ , the support lies in $B (0, 2)$ .

2.1F2step 1.1

Let $γ : [0, 1] \to R^{2}$ be continuous with $∣ γ (0) ∣ = 1 / 2$ and $∣ γ (1) ∣ = 2$ . The scalar map $t \mapsto ∣ γ (t) ∣$ is continuous because $∣ ∣ x ∣ - ∣ y ∣ ∣ \leq ∣ x - y ∣$ . By [F2], it equals $1$ for some $t$ , so $γ (t) \in S^{1}$ and the support separates the two spheres.

3.1F1given∎

The support of $δ_{0}$ is ${0}$ by [F1]. The radial segment $γ (t) = ((1 - t) / 2 + 2 t, 0)$ joins $(1 / 2, 0)$ to $(2, 0)$ and has strictly positive norm throughout, so it avoids ${0}$ . Thus this support does not separate the spheres. No conclusion about energy minimizers follows from either topological calculation.

Depends on

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Sources

Lénaïc Chizat and Francis Bach, On the Global Convergence of Gradient Descent for Over-parameterized Models using Optimal Transport (standard reference, not scraped)
Ambrosio, Gigli, and Savaré, Gradient Flows in Metric Spaces and in the Space of Probability Measures (standard reference, not scraped)