Pseudo centre and its applications

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Abstract

This paper introduces the notion of a ‘pseudo centre’ of a suitable bounded region in Rn. We investigate general properties of a pseudo centre by means of techniques provided by calculus, differential geometry, and topology. While calculus allows to introduce ‘depth diagram’ and relates that to constraint extreme value problems, the topological point of view allows to study the qualitative properties of a pseudo centre. We have discussed the relation of index of a pseudo centre to Lusternik-Schnirelmann category of a space. The paper is concluded by some discussions regarding the generalised version of the problem and further research.

Introduction

Geometric properties of solid objects including shape and size are important mathematical concepts in practical fields; see [6] for such issues in food processing industry. It has also application in computer science [12], image processing and pattern recognitions, and analysis of biomedical images [2]. Geometric properties of regions is also important in other fields; for instance, in the art gallery problem [11]. Here, we consider some geometric properties of shapes. In the plane, a symmetric object like a disk has a centre with the property that any point on the boundary of the disk can be connected to the centre with an open line segment which entirely belongs to the interior of the disk. Similar to the disk, this property also holds for other bounded convex regions. However, this property does not always hold for an arbitrary bounded region in Rn. For example, the planar region consisting of points between two circles with the same centre but different radii does not have a centre in the above sense. In this case the boundary of the considered region is not connected. It is also easy to build a region without this property while having connected boundary. Paving the boundary with bounded open line segments which are entirely located in the interior of the given planar region motivates the main objective of this paper. If it is not possible to pave the boundary by bounded open line segments through a single point, one might do this job with a bigger subset of the interior of the given region. This motivates the concepts of pseudo centre for planar regions. Such investigation has also application in visualisation of constraint extreme value problems. The idea can be easily extended to higher dimensions.

Organisation of the paper. In Section 2, we introduce the notion of pseudo centre and discuss some of its natural properties. In Section 3 we study the behaviour of pseudo centre in R2 under diffeomorphisms. We introduce the notion of induced partition of the boundary in Section 5. Section 6 introduces the notion of depth diagram followed by the related visualization technique. In this paper, we only offer some basic and try to set up the mathematical framework. Consequently, in Section 7 offers a collection of discussions and problems that might be of interest and need an independent investigation.

Section snippets

Pseudo centre and its index

We begin with fixing some terminology. For two manifolds M and N, M≅N means M is diffeomorphic to N. For any pair of (q,p)∈Rn we denote the closed line segment {λp+(1−λ)q}|λ∈[0,1] by Lq(p) and the half open line segment {λp+(1−λ)q}|λ∈[0,1) by Lq0(p). For M⊆Rn, ∂M⊆Rn is the topological boundary that is ∂M=clRn(M)∩clRn(Rn−M) where clRn denotes the closure operation with respect to the topology of Rn. We also use int for the interior point operator given with int(M)=M−∂(M). The standard n-ball in R

Pseudo centre and diffeomorphisms

We wish to study the behaviour of a pseudo centre under smooth transformations Rn→Rn which act as diffeomorphisms on a given region D⊆Rn to which we have a given pseudo centre. The following is immediate and evident, so we omit its proof.

Lemma 3.1

Suppose ϕ:Rn→Rn is a conformal map, D⊆Rn, and Q={q1,…,qs} is a pseudo centre for D. Then ϕ(Q) is a pseudo centre for ϕ(D). Moreover, D is of least index if and only if ϕ(D) is.

We wish to study to the case of regions in the plane and set n=2. In this case, due

Towards a cobordism invariant

Theorem 3.4 as its proof demonstrates an interesting point about pseudo centre of a given space M and its behaviour under continuous/smooth deformations. It might seem that such deformations are always possible, and one can transform a region so that its pseudo centre behaves in a prescribed way. In this section, using the tools of cobordism, we show that this is a false claim, at least if we restrict ourselves to one specific type transformations.

The motivating example. Indeed, one of

Induced partitions of the boundary

Let D be a connected compact subset of Rn with (n−1)-manifold boundary ∂D. A k-partition of ∂D is represented as B={Bi}i=1k where Bi⊂∂D is connected for each i, we have ∂D=∪i=1kBi, and for any i≠j we have dim⁡(Bi∩Bj)≤n−2. A k-partition may be called a partition when k is known and fixed. For a given set D, it is worth to note that k-partition is not necessarily unique.

Lemma 5.1

Any pseudo centre Q for D induces a partition B for ∂D such that for each B∈B there is q∈Q such that B has a pseudo centre with

Depth diagram

When dealing with a problem on a boundary, one issue is the function over the boundary. We can view the boundary as a closed curve and model that as a constraint. Therefore, we have a function on the plane with constraint. The variations of function and its extreme points can tell us about the variations of the function on boundary. This is related to the constraint extreme value problem which is perfectly studied in literature. However, a good visualization of such function, its variations,

Open problems

We presented the concept of pseudo centre and depth diagram. The presented methodology is applicable to real world related problems. It is also clear that in real applications of the theory, one may face some computational complexity. Apart from the part of the carried work, there are many steps which can be taken to improve this idea in real world applications. Here, we discuss some of these steps.

The problem of pseudo centre, its minimal form and its applications are subject to computations.

Acknowledgements

We are grateful to Mark Grant who read an earlier version of this paper and pointed out that the similar (dual) problem of the ‘Art Gallery Problem’ is studied in the field of computer sciences.

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