Characterization of Cheeger sets for convex subsets of the plane

Abstract

Given a planar convex domain , its Cheeger set Ꮿ is defined as the unique minimizer of |∂ X|/|X| among all nonempty open and simply connected subsets X of . We prove an interesting geometric property of Ꮿ , characterize domains which coincide with Ꮿ and provide a constructive algorithm for the determination of Ꮿ .

An Introduction to Total Variation for Image Analysis

“…From the Euler-Lagrange equation (see also (26)) it follows that the curvature of ∂E s ∩ C is (1 − s)/λ. An accurate study (see also [5,4,42]) shows that, if we define (for…”

“…Compare with (42). This is an essential property, as numerous algorithms have been designed (mostly in the 70's) to find the zeroes of monotone operators and their numerous variants.…”

Two dimensions are easier

“…When Ω is a square, the corresponding Cheeger set is a rounded square and can be calculated by elementary means [13]. Incidentally, the Cheeger constant 2 .…”

Anisotropic mean curvature on facets and relations with capillarity

“…Concerning uniqueness, examples of planar sets Ω admitting more then one (Euclidean) Cheeger subset, and also an uncountable family of Cheeger subsets, can be found in [55], [58]. (17) Further results hold for a convex Ω ⊂ R m , see [4].…”

“…Finally we have a complete characterization of the (unique) Cheeger subset of a planar convex domain, proven in [55] for the Euclidean norm and in [56] for a general anisotropy. (20) …”