Quantities of positive reach

Amounts of positive coverage ( Engl. : Sets with positive reach) are in the geometry of a class of subsets of Euclidean space (or more generally Riemannian manifolds ) that the concept of convex sets generalize. They were introduced in 1959 by the American mathematician Herbert Federer . Quantities of positive range have found widespread use , especially in geometric dimension theory and curvature theory . They are able to model real objects more flexibly than, for example, differentiable manifolds and yet simple enough to be accessible to analytical methods.

Definitions

Let be a subset of a Euclidean space. ${\displaystyle A\subseteq \mathbb {R} ^{n}}$

Hint: Some authors assume here a non-empty subset of a smooth, connected Riemannian manifold.

Let the corresponding distance function be further , where the Euclidean norm denotes. ${\displaystyle d_{A}(x):=\inf _{a\in A}\|x-a\|_{2}}$ ${\displaystyle \|.\|_{2}}$

Building on this, the following terms can now be formulated:

Definitely the next point

With

${\displaystyle Unp(A):=\{x\in \mathbb {R} ^{n}|\exists !\ a^{*}\in A:d_{A}(x)=\|x-a^{*}\|_{2}\}}$

is the set of all clearly nearest points of designated (of English :. un ique Closest p oints). The quantifier means existence and uniqueness of the next point in . ${\displaystyle A}$ ${\displaystyle \exists !}$ ${\displaystyle A}$

It is easy to see that must always apply. ${\displaystyle A\subseteq Unp(A)}$

The canonical surjection is called the metric projection on . Restricted to them is the identity . ${\displaystyle \Pi _{A}\colon Unp(A)\to A\ ;\ x\mapsto a^{*}}$ ${\displaystyle A}$ ${\displaystyle A}$

Range of a point

It is for a point and the open ball around with radius . Then be for a point ${\displaystyle x\in \mathbb {R} ^{n}}$ ${\displaystyle \varepsilon >0}$ ${\displaystyle B^{\circ }(x;\varepsilon )=\{y\in \mathbb {R} ^{n}|\|x-y\|_{2}<\varepsilon \}}$ ${\displaystyle x}$ ${\displaystyle \varepsilon }$ ${\displaystyle a\in A}$

${\displaystyle reach(A;a):=\sup\{r\geq 0|B^{\circ }(a;r)\subseteq Unp(A)\}}$

the reach of this point.

Range of a crowd

The above definition can be applied naturally to sets, so be

${\displaystyle reach\ A:=\inf _{a\in A}\ reach(A;a)}$

the range of . ${\displaystyle A}$

There is a vivid explanation of this term: if it has a lot of positive reach, its edge is smooth enough to allow a radiused ball to roll along it. ${\displaystyle A}$ ${\displaystyle \partial A}$ ${\displaystyle reach\ A}$

properties

• Sets with positive range are necessarily closed , that is, the mentioned margin is included in the set.
• A set has infinite range if and only if it is closed and convex.
  • In particular, a convex (closed) set has a positive range.
• A compact, contiguous submanifold of Euclidean space has positive range. ${\displaystyle {\mathcal {C}}^{2}}$
• For any set , the distance function is Lipschitz continuous with constant 1. ${\displaystyle A}$ ${\displaystyle d_{A}}$
• In addition, the association is constantly on . ${\displaystyle a\mapsto reach(A;a)}$ ${\displaystyle A}$
• If there is also a positive range, the metric projection onto is also continuous for each Lipschitz. ${\displaystyle A}$ ${\displaystyle \Pi _{A}}$ ${\displaystyle A_{r}=\{x\in \mathbb {R} ^{n}|d_{A}(x)\leq r\}}$ ${\displaystyle 0\leq r<reach\ A}$

Individual evidence

1. ^ Herbert Federer, Curvature Measures , Transactions of the American Mathematical Society 93, 418-491, 1959
2. ↑ Christoph Thäle, Singular Curvature Theory , guest lecture at Ulm University, memory protocol, May 28, 2008
3. ^ Victor Bangert, Sets with positive reach ; in: Archiv der Mathematik 38/1, 54–57, 1982; Quoted from: http://link.springer.com/10.1007%2FBF01304757?from=SL Accessed June 25, 2012
4. ↑ Christoph Thäle, 50 Years sets of positive reach - A survey ; in: Surveys in Mathematics and its Applications Vol. 3, 123-165, 2008; Quoted from: http://www.kurims.kyoto-u.ac.jp/EMIS/journals/SMA/v03/v03.html Accessed June 25, 2012