DefinitionDefinition: Literature-sourcedProof: Not applicablePipeline-generatedjudge pass (gpt-6-luna)verified 2026-10-22 (gpt-6.1-sol)

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Variational curvature of a finite-perimeter set

Definition

Let $Ω = R^{2}$ or let $Ω \subseteq R^{2}$ be a bounded Lipschitz domain. Choose one of three perimeter functionals for measurable $E \subseteq Ω$ : full-space perimeter $P_{R} (E) = P (E; R^{2})$ when $Ω = R^{2}$ ; Neumann relative perimeter $P_{N} (E) = P (E; Ω)$ on a bounded domain; or Dirichlet perimeter $P_{D} (E) = P (\tilde{E}; R^{2})$ , where $\tilde{E}$ is $E$ extended by the empty set outside $Ω$ . Require finite measure and finite selected perimeter, and let $κ : Ω \to R$ be Lebesgue-measurable with $\int_{F} ∣ κ ∣ < \infty$ for every set considered. The set $E$ has variational curvature $κ$ for the selected model if its selected perimeter minus $\int_{E} κ d x$ is no greater than the same quantity for every finite-measure finite-selected-perimeter competitor $F \subseteq Ω$ whose symmetric difference $F △ E$ has compact closure in $Ω$ . In the full-space model, this means compact closure in $R^{2}$ .

For bounded Dirichlet TV, a separate obstacle curvature condition on $Ω$ means that $Ω$ , viewed as a finite-perimeter set in $R^{2}$ , minimizes $F \mapsto P (F; R^{2}) - \int_{F} κ_{Ω} d x$ over all finite-measure finite-perimeter $F \subseteq R^{2}$ whose symmetric difference with $Ω$ has compact closure in $R^{2}$ , where $κ_{Ω} \in L^{1} (R^{2}; R)$ .

Definition references

Relative and full-space perimeters are TV of characteristic functions.

Dirichlet perimeter uses zero extension; Neumann perimeter is relative to the open domain.

Iglesias, Mercier, and Scherzer, A note on convergence of solutions of total variation regularized linear inverse problems, Definition 4

Chambolle, Duval, Peyré, and Poon, Geometric properties of solutions to the total variation denoising problem, §2.6

Depends on

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Sources

José A. Iglesias, Gwenael Mercier, and Otmar Scherzer, A note on convergence of solutions of total variation regularized linear inverse problems, Definition 4 (standard reference, not scraped)
Antonin Chambolle, Vincent Duval, Gabriel Peyré, and Clarice Poon, Geometric properties of solutions to the total variation denoising problem, §2.6 (standard reference, not scraped)