The Cheeger constant of curved tubes

Abstract

We compute the Cheeger constant of spherical shells and tubular neighbourhoods of complete curves in an arbitrary dimensional Euclidean space.

On the Cheeger problem for rotationally invariant domains

“…We anticipate that the existence and, moreover, uniqueness of such Cheeger set holds true provided Ω is generated by a closed, convex curve Γ : [𝛼, 𝛽] → R×(0, +∞). In some particular cases, such as the torus in R 𝑛 , this has been proven in [17].…”

“…On the other hand, in the higher dimensional case 𝑛 ≥ 3, there is a big variety of CMC surfaces, and a characterization of 𝐶 by "rolling" a ball inside Ω as in (1.3) can be violated, see [14,Remark 13]. Moreover, an explicit characterization of Cheeger sets seems to be known only for some particular domains such as ellipsoids of low eccentricity [2], spherical shells [8], and tubular neighbourhoods of complete curves [17], while even for three-dimensional cubes the problem is open [13].…”

Cheeger Sets for Rotationally Symmetric Planar Convex Bodies

“…Some references which cover quite well this case are [31,[33][34][35][36]47]. In higher dimensions, there are not so many references, and the reader may check [13,15,30]. Recall that the uniqueness of the Cheeger set is not assured in this non-convex setting, as shown in [27] by describing a particular example.…”

Quantitative lower bounds to the Euclidean and the Gaussian Cheeger constants

“…It is of interest to provide estimates on the constants as there are no available formulas to directly compute them, but for very few classes of sets Ω limited to the Euclidean 2-dimensional setting, see [36,39,40,42,49]. In higher dimensions, some very special cases are treated in [3,8,17,37]. To give an upper bound to h(Ω) (resp., h γ (Ω)) it is enough to compute the ratio P (E)/|E| (resp., P γ (E)/γ(E)) for any competitor E, while establishing lower bounds exploits the relevant isoperimetric inequalities.…”