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The central set of a domain D is the set of centers of maximal discs in D. Fremlin showed in (3) that the central set of a planar domain has zero area and asked whether it can have Hausdorff dimension strictly larger than 1. We construct a planar domain with central set of Hausdorff dimension 2.
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Citations
... The medial axis is a subset of the central set of Ω, which is the set of the centers of the maximal balls contained in Ω. In [5] Bishop and Habokyan prove that this set can have Hausdorff dimension arbitrarily close to 2, even though the medial axis can have Hausdorff dimension at most 1. ...
We show that every Jordan quadrilateral $Q\subset\mathbb{C}$ contains a disk $D$ so that $\partial D\cap\partial Q$ contains points of three different sides of $Q$. As a consequence, together with some modulus estimates from Lehto and Virtanen, we offer a short proof of the main result obtained by Chrontsios-Garitsis and Hinkkanen in 2024 and it also improves the bounds on their result.
... In view of these simple examples, the second equation in (51) (which holds for every ρ > 0!) is quite surprising since Cut φ (A) ∩ Unp φ (A) can be a much larger set than R n+1 \ (A ∪ Unp φ (A)). In fact, since R n+1 \ (A ∪ Unp φ (A)) is always an n-dimensional set (see section 2.4), it follows from the example in [BH08] that the set Cut φ (A) ∩ Unp φ (A) can be an (n + 1)-dimensional set! Lemma 4.2. ...
Extending the celebrated results of Alexandrov (1958) and Korevaar-Ros (1988) for smooth sets, as well as the results of Schneider (1979) and the first author (1999) for arbitrary convex bodies, we obtain for the first time the characterization of the isoperimetric sets of a uniformly convex smooth finite-dimensional normed space (i.e. Wulff shapes) in the non-smooth and non-convex setting, based on the natural geometric condition involving the curvature measures. More specifically we show, under a natural mean-convexity assumption, that finite unions of disjoint Wulff shapes are the only sets of positive reach $ A \subseteq \mathbf{R}^{n+1} $ with finite and positive volume such that, for some $ k \in \{0, \ldots , n-1\}$, the $ k $-th generalized curvature measure $ \Theta^\phi_{k}(A, \cdot) $, which is defined on the unit normal bundle of $ A $ with respect to the relative geometry induced by $ \phi $, is proportional to $ \Theta^\phi_{n}(A, \cdot)$. If $ k = n-1 $ the conclusion holds for all sets of positive reach with finite and positive volume. We also prove a related sharp result about the removability of the singularities. This result is based on the extension of the notion of a normal boundary point, originally introduced by Busemann and Feller (1936) for arbitrary convex bodies, to sets of positive reach. These results are new even in the Euclidean space. Several auxiliary and related results are also proved, which are of independent interest. They include the extension of the classical Steiner--Weyl tube formula to arbitrary closed sets in a finite dimensional uniformly convex normed vector space and a general formula for the derivative of the localized volume function, which extends and complements recent results of Chambolle--Lussardi--Villa (2021).
... For polygons the two sets are the same, but in general they are not (e.g., the parabolic region Ω = {(x, y) : y > x 2 } contains a maximal disk that is only tangent at the origin). More dramatically, the medial axis of a planar domain always has σ-finite 1-dimensional measure [62], but the central set can have Hausdorff dimension 2, [15]. Some papers in the mathematical literature that deal with the medial axis include [6], [25], [50], [59], [68], [69], [80], [94], [95], [123]. ...
Given any $\epsilon >0$ and any planar region $\Omega$ bounded by a simple n-gon $P$ we construct a ($1 + \epsilon)$-quasiconformal map between $\Omega$ and the unit disk in time $C(\epsilon)n$. One can take $ C(\epsilon) = C + C \log (1/\epsilon) \log \log (1/\epsilon)$.
... There are domains whose skeletons demonstrate irregular behavior (see, e.g., [7]). However, this is not the case for domains that appear in applications. ...
The aim of this note is to extend the notion of Fréchet distance over the set of weighted rooted trees. The weighted trees naturally appear as skeletons of planar domains. The defined distance allows for defining a distance between (weighted) threes, which is merely a symmetric, i.e., does not necessarily satisfy the triangle inequality.
... All these sets, which have each one its own role in the geometry of the distance function from the boundary, have been widely investigated in the literature, often with a non-uniform terminology. A miscellaneous collection of related references, without any attempt of completeness, is [1,3,9,27,32,33,40,41]. It must be added that recently the singular set of the distance function has raised an increasing interest also in applied domains, such as computer science and visual reconstruction, and this is especially true for the central set (often named medial axis in this context), see e.g. ...
... Then, given x ∈ S, there exists a sequence {x h } contained into S r \ S, with lim h x h = x. By applying (5) to each x h , and then passing to the limit as h → +∞, we get d (Sr ) c (x) = r, which extends the validity of (5) to S and proves (9). In view of (9), it is clear that S = M(S r ) = C(S r ); then (7) follows recalling (2) and the fact that S is closed. ...
... We remark that the property of Σ(Ω) and M (Ω) of having null Lebesgue measure is not enjoyed in general by Σ(Ω): in [41, Section 3], there is an example of two-dimensional convex set Ω whose cut locus has positive Lebesgue measure. We also point out that the central set C(Ω) of a planar domain may fail to be H 1 -rectifiable (see the examples in [32, Section 4]), and it may even happen to have Hausdorff dimension 2 (see[9]). ...
We give a complete characterization, as "stadium-like domains", of convex
subsets $\Omega$ of $\mathbb{R}^n$ where a solution exists to Serrin-type
overdetermined boundary value problems in which the operator is either the
infinity Laplacian or its normalized version. In case of the not-normalized
operator, our results extend those obtained in a previous work, where the
problem was solved under some geometrical restrictions on $\Omega$. In case of
the normalized operator, we also show that stadium-like domains are precisely
the unique convex sets in $\mathbb{R}^n$ where the solution to a Dirichlet
problem is of class $C^{1,1} (\Omega)$.
... All these sets, which have each one its own role in the geometry of the distance function from the boundary, have been widely investigated in the literature, often with a non-uniform terminology. A miscellaneous collection of related references, without any attempt of completeness, is [1,3,9,26,30,31,38,39]. It must be added that recently the singular set of the distance function has raised an increasing interest also in applied domains, such as computer science and visual reconstruction, and this is especially true for the central set (often named medial axis in this context), see e.g. ...
... We remark that the property of Σ(Ω) and M (Ω) of having null Lebesgue measure is not enjoyed in general by Σ(Ω): in [39, Section 3], there is an example of two-dimensional convex set Ω whose cut locus has positive Lebesgue measure. We also point out that the central set C(Ω) of a planar domain may fail to be H 1 -rectifiable (see the examples in [30, Section 4]), and it may even happen to have Hausdorff dimension 2 (see [9]). ...
... We have to prove that [∂B r (p) ∩ Σ + ] ⊆ C r (p) . We claim that there exists δ ∈ (0, r) such that (9) π S ∂S r ∩ Σ + ∩ B δ (a) ⊆ {p} . ...
We provide a complete characterization of closed sets with empty interior and
positive reach in $\mathbb{R}^2$. As a consequence, we characterize open
bounded domains in $\mathbb{R}^2$ whose high ridge and cut locus agree, and
hence $C^1$ planar domains whose normal distance to the cut locus is constant
along the boundary. The latter results extends to convex domains in
$\mathbb{R}^n$.
... For polygons the two sets are the same, but in general they are not (e.g., the parabolic region Ω = {(x, y) : y > x 2 } contains a maximal disk which is only tangent at the origin). More dramatically, the medial axis of a planar domain always has σ-finite 1-dimensional measure [69], but the central set can have Hausdorff dimension 2, [19]. ...
Given any ǫ > 0 and any planar region bounded by a simple n-gon P we construct a (1 + ǫ)-quasiconformal map between and the unit disk in time C(ǫ)n. One can take C(ǫ) = C + C log 1 ǫ loglog 1 ǫ .
... A point which is not a vertex is called an interior edge point (and the corresponding disk hits ∂D in exactly two points). The medial axis is always a union of countably many rectifiable arcs (see [6]; also [4], [5]). In particular, Fremlin shows in [6] that there is a λ < ∞, so that for each point z in the medial axis there is a ball B(z, r) so that any point w ∈ B(z, r) can be connected to z by a path of length ≤ λδ. ...
We show that any simply connected rectifiable domain Ω can be de-composed into Lipschitz crescents using only crosscuts of the domain and using total length bounded by a multiple of the length of ∂Ω. In particular, this gives a new proof of a theorem of Peter Jones that such a domain can be decomposed into Lipschitz disks.
For the distance function from any closed subset of any complete Finsler manifold, we prove that the singular set is equal to a countable union of delta‐convex hypersurfaces up to an exceptional set of codimension two. In addition, in dimension two, the whole singular set is equal to a countable union of delta‐convex Jordan arcs up to isolated points. These results are new even in the standard Euclidean space and shown to be optimal in view of regularity.
Extending the celebrated results of Alexandrov (1958) and Korevaar–Ros (1988) for smooth sets, as well as the results of Schneider (1979) and the first author (1999) for arbitrary convex bodies, we obtain for the first time the characterization of the isoperimetric sets of a uniformly convex smooth finite-dimensional normed space (i.e. Wulff shapes) in the non-smooth and non-convex setting, based on a natural geometric condition involving the curvature measures. More specifically we show, under a weak mean convexity assumption, that finite unions of disjoint Wulff shapes are the only sets of positive reach A⊆Rn+1 with finite and positive volume such that, for some k∈{0,…,n−1}, the k-th generalized curvature measure Θkϕ(A,⋅), which is defined on the unit normal bundle of A with respect to the relative geometry induced by ϕ, is proportional to the top order generalized curvature measure Θnϕ(A,⋅). If k=n−1 the conclusion holds for all sets of positive reach with finite and positive volume. We also prove a related sharp result about the removability of the singularities. This result is based on the extension of the notion of a normal boundary point, originally introduced by Busemann and Feller (1936) for arbitrary convex bodies, to sets of positive reach.
These findings are new even in the Euclidean space.
Several auxiliary and related results are proved, which are of independent interest. They include the extension of the classical Steiner–Weyl tube formula to arbitrary closed sets in a finite dimensional uniformly convex normed vector space, a general formula for the derivative of the localized volume function, which extends and complements recent results of Chambolle–Lussardi–Villa (2021), and general versions of the Heintze–Karcher inequality.
