Medial axis and singularities. (English) Zbl 1380.32009
This comprehensive paper is devoted to the study of the medial axis and other related sets associated with closed subsets \(X\) of \(\mathbb R^n\) definable in polynomially bounded o-minimal structures. Many of the results also hold without definability assumption. The focus lies on the connection of the above-mentioned sets to the singularities of \(X\).
The medial axis \(M_X\) is the set of points \(x\) with more than one closest point in \(X\). The central set \(C_X\) is the set of centers of maximal balls contained in \(\mathbb R^n \setminus X\). Section 2 in the paper presents a detailed study of these sets (among others) as well as of the properties of the related set-valued functions \(\mathbb R^n \ni x \mapsto m(x) := \{y \in X : \|x-y\| = d(x,X)\}\) and \(X \ni a \mapsto N(a) :=\{x \in \mathbb R^n : a \in m(x)\}\). Furthermore, the connection of \(M_X\) to the singularities (i.e., non-differentiability points) of the function \(\delta(x) := d(x,X)^2\) is explicated.
Section 3 is concerned with the question when the medial axis reaches the singularities of hypersurfaces. For \(k \in \mathbb N \cup \{\infty,\omega\}\) let \(\text{Reg}_k X := \{x \in X : X \text{ is a } C^k \text{ submanifold at } x\}\) and \(\text{Sng}_k X := X \setminus \text{Reg}_k X\). A complete answer is given in the plane case when \(X\) is a pure one-dimensional closed set in \(\mathbb R^2\) definable in a polynomially bounded o-minimal structure. If \(0 \in \text{Reg}_1 X \cap \text{Sgn}_2 X\), then \(0 \in \overline {M_X}\) if and only if \(X\) is superquadratic at \(0\). The latter means that, near \(0\), \(X\) is the graph of a function germ of order \(<2\). If \(0 \in \text{Sgn}_1 X\) and the germ \((X \setminus \{0\},0)\) has at least two connected components, then \(0 \in \overline {M_X}\). If the germ \((X \setminus \{0\},0)\) is connected, then \(0 \not\in \overline {M_X}\). If \(0 \in \text{Reg}_2 X\), then \(0 \not\in \overline {M_X}\). Furthermore, a description of the Peano tangent cone \(C_0(M_X)\) of \(M_X\) at \(0\) is given in the case that \(M_X\) reaches \(X\).
Section 4 deals with hypersurfaces \(X\) in \(\mathbb R^n\). First, it is proved that \(C_0(M_X) \supseteq \overline {M_{C_0(X)}}\) provided that \(C_0(X)\) has empty interior and is not contained in a hyperplane. For \(C^1\) hypersurfaces \(X\) the relation between the local Lipschitz constant of a continuous unit normal vector field at \(a \in X\), the local reach radius at \(a\), and the condition \(a \in \overline {M_X}\) is clarified. Finally, for points \(a\) in a closed nonempty subset \(X \subseteq \mathbb R^n\) definable in a polynomially bounded o-minimal structure a notion of definable reaching radius \(r(a)\) is developed. It is proved that \(r(a) = 0\) if and only if \(a \in \overline {M_X}\).
The medial axis \(M_X\) is the set of points \(x\) with more than one closest point in \(X\). The central set \(C_X\) is the set of centers of maximal balls contained in \(\mathbb R^n \setminus X\). Section 2 in the paper presents a detailed study of these sets (among others) as well as of the properties of the related set-valued functions \(\mathbb R^n \ni x \mapsto m(x) := \{y \in X : \|x-y\| = d(x,X)\}\) and \(X \ni a \mapsto N(a) :=\{x \in \mathbb R^n : a \in m(x)\}\). Furthermore, the connection of \(M_X\) to the singularities (i.e., non-differentiability points) of the function \(\delta(x) := d(x,X)^2\) is explicated.
Section 3 is concerned with the question when the medial axis reaches the singularities of hypersurfaces. For \(k \in \mathbb N \cup \{\infty,\omega\}\) let \(\text{Reg}_k X := \{x \in X : X \text{ is a } C^k \text{ submanifold at } x\}\) and \(\text{Sng}_k X := X \setminus \text{Reg}_k X\). A complete answer is given in the plane case when \(X\) is a pure one-dimensional closed set in \(\mathbb R^2\) definable in a polynomially bounded o-minimal structure. If \(0 \in \text{Reg}_1 X \cap \text{Sgn}_2 X\), then \(0 \in \overline {M_X}\) if and only if \(X\) is superquadratic at \(0\). The latter means that, near \(0\), \(X\) is the graph of a function germ of order \(<2\). If \(0 \in \text{Sgn}_1 X\) and the germ \((X \setminus \{0\},0)\) has at least two connected components, then \(0 \in \overline {M_X}\). If the germ \((X \setminus \{0\},0)\) is connected, then \(0 \not\in \overline {M_X}\). If \(0 \in \text{Reg}_2 X\), then \(0 \not\in \overline {M_X}\). Furthermore, a description of the Peano tangent cone \(C_0(M_X)\) of \(M_X\) at \(0\) is given in the case that \(M_X\) reaches \(X\).
Section 4 deals with hypersurfaces \(X\) in \(\mathbb R^n\). First, it is proved that \(C_0(M_X) \supseteq \overline {M_{C_0(X)}}\) provided that \(C_0(X)\) has empty interior and is not contained in a hyperplane. For \(C^1\) hypersurfaces \(X\) the relation between the local Lipschitz constant of a continuous unit normal vector field at \(a \in X\), the local reach radius at \(a\), and the condition \(a \in \overline {M_X}\) is clarified. Finally, for points \(a\) in a closed nonempty subset \(X \subseteq \mathbb R^n\) definable in a polynomially bounded o-minimal structure a notion of definable reaching radius \(r(a)\) is developed. It is proved that \(r(a) = 0\) if and only if \(a \in \overline {M_X}\).
Reviewer: Armin Rainer (Wien)
MSC:
| 32B20 | Semi-analytic sets, subanalytic sets, and generalizations |
| 54F99 | Special properties of topological spaces |
| 03C64 | Model theory of ordered structures; o-minimality |
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