aa r X i v : . [ m a t h . M G ] N ov PROXIMAL VORONO¨I REGIONS
J.F. PETERS
Dedicated to the Memory of Som Naimpally
Abstract.
A main result in this paper is the proof that proximal Vorono¨ıregions are convex polygons. In addition, it is proved that every collection ofproximal Vorono¨ı regions has a Leader uniform topology. Introduction
Klee-Phelps convexity [8, 12] and related results [11] are viewed here in terms ofVorono¨ı regions, named after the Ukrainian mathematician Georgy Vorono¨ı [13, 14,15]. A nonempty set A of a space X is a convex set , provided αA + (1 − α ) A ⊂ A for each α ∈ [0 ,
1] [1, § simple convex set is a closed half plane (allpoints on or on one side of a line in R ). Lemma 1.1. [6, § The intersection of convex sets is convex.Proof.
Let
A, B ⊂ R be convex sets and let K = A ∩ B . For every pair points x, y ∈ K , the line segment xy connecting x and y belongs to K , since this propertyholds for all points in A and B . Hence, K is convex. (cid:3) pV p Figure 1. V p = Intersection of closed half-planesLet S ⊂ R be a finite set of n points called sites, p ∈ S . The set S is called the generating set [7]. Let H pq be the closed half plane of points at least as close to p as to q ∈ S \ { p } , defined by H pq = (cid:26) x ∈ R : k x − p k ≤ q ∈ S k x − q k (cid:27) . Mathematics Subject Classification.
Primary 65D18; Secondary 54E05, 52C20, 52C22.
Key words and phrases.
Convex polygon, proximal, Leader uniform topology, Vorono¨ı region.The research has been supported by the Scientific and Technological Research Council ofTurkey (T ¨UB˙ITAK) Scientific Human Resources Development (BIDEB) under grant no: 2221-1059B211402463 and Natural Sciences & Engineering Research Council of Canada (NSERC) dis-covery grant 185986. A convex polygon is the intersection of finitely many half-planes [5, § I.1, p. 2]. See,for example, Fig. 1.
Remark 1.2.
The Vorono¨ı region V p depicted as the intersection of finitely manyclosed half planes in Fig. 1 is a variation of the representation of a Vorono¨ı regionin the monograph by H. Edelsbrunner [6, § p and perpendicularto the sides of V p are comparable to the lines leading from the center of the convexpolygon in G.L. Dirichlet’s drawing [3, §
3, p. 216]. (cid:4) Preliminaries
Let S ⊂ E , a finite-dimensional normed linear space. Elements of S are calledsites to distinguish them from other points in E [6, § p ∈ S . A Vorono¨ı region of p ∈ S (denoted V p ) is defined by V p = (cid:26) x ∈ E : k x − p k ≤ ∀ q ∈ S k x − q k (cid:27) . Remark 2.1.
A Vorono¨ı region of a site p ∈ S contains every point in the planethat is closer to p than to any other site in S [7, § V p , V q be Vorono¨ıpolygons. If V p ∩ V q is a line, ray or line segment, then it is called a Vorono¨ı edge .If the intersection of three or more Vorono¨ı regions is a point, that point is calleda
Vorono¨ı vertex . (cid:4) Lemma 2.2.
A Vorono¨ı region of a point is the intersection of closed half planesand each region is a convex polygon.Proof.
From the definition of a closed half-plane H pq = (cid:26) x ∈ R : k x − p k ≤ q ∈ S k x − q k (cid:27) ,V p is the intersection of closed half-planes H pq , for all q ∈ S − { p } [5], forming apolygon. From Lemma 1.1, V p is a convex. (cid:3) A Voronoi diagram of S (denoted by V ) is the set of Voronoi regions, one foreach site p ∈ S , defined by V = [ p ∈ S V p . Example 2.3. Centroids as Sites in an Image Tessellation .Let E be a segmentation of a digital image and let S ⊂ E be a set of sites, whereeach site is the centroid of a segment in E . In a centroidal approach to the Vorono¨ıtessellation of E , a Vorono¨ı region V p is defined by the intersection of closed halfplains determined by centroid p ∈ S . The centroidal approach to Voronoi tessella-tion was introduced by Q. Du, V. Faber, M. Gunzburger [4] . (cid:4) Main Results
Let V p , V z be Vorono¨ı regions of p, z ∈ S , a set of Vorono¨ı sites in a finite-dimensional normed linear Space E that is topological, cl A the closure of a nonemptyset A in E . V p , V z are proximal (denoted by V p δ V z ), provided P = cl V p ∩ cl V z = ∅ [2]. The set P is called a proximal Vorono¨ı region . Theorem 3.1.
Proximal Vorono¨ı regions are convex polygons.
ROXIMAL VORONOI REGIONS 3
Proof.
Let P be a proximal Vorono¨ı region. By definition, P is the nonemptyintersection of convex sets. From Lemma 1.1, P is convex. Consequently, P is theintersection of finitely many closed half planes. Hence, from Lemma 2.2, P is aVorono¨ı region of a point and is a convex polygon. (cid:3) Corollary 3.2.
The intersection of proximal Vorono¨ı regions is either a Vorono¨ıedge or Vorono¨ı point.
Any two Vorono¨ı regions intersect at least a vertex and at most along theirboundaries. Together, the set of Vorono¨ı regions V cover the entire plane [5, § S ⊂ E , a Vorono¨ı diagram D of S is the set of Voronoiregions, one for each site in S . Corollary 3.3.
A Vorono¨ı diagram D equals V . The partition of a plane E with a finite set of n sites into n Vorono¨ı polygons isknown as a Dirichlet tessellation, named after G.L. Dirichlet [16] (see [3]). A cover (covering) of a space X is a collection U of subsets of X whose union contains X ( i.e. , U ⊇ X ) [17, § § Corollary 3.4.
A Dirichlet tessellation D of the Euclidean plane E is a coveringof E . Recall that the Euclidean space E = R is a metric space. The topology ina metric space results from determining which points are close to each set in thespace. A point x ∈ E is close to A ⊂ E , provided the Hausdorff distance d ( x, A ) = inf {k x − a k : a ∈ A } = 0. Let X, Y be a pair of metric spaces, f : X −→ Y isa function such that for each x ∈ X , there is a unique f ( x ) ∈ Y . A continuousfunction preserves the closeness (proximity) between points and sets, i.e. , f ( x ) isclose to f ( B ) whenever x is close to B . In a proximity space, one set A is nearanother set B , provided A δ B , i.e. , the closure of A has at least one elementin common with the closure of B . The set A is close to the set B , provided the˘Cech distance D ( A, B ) = inf {k a − b k : a ∈ A, b ∈ B } = 0. In that case, we write A δ B ( A and B are proximal). A uniformly continuous mapping is a function thatpreserves proximity between sets, i.e. , f ( A ) δ f ( B ) whenever A δ B . A
Leaderuniform topology is determined by finding those points that are close to each givenset in E . Theorem 3.5.
Let S be a set of two or more sites, p ∈ S, V p ∈ D in the Euclideanspace R . Then1 o V p is near at least one other Vorono¨ı region in D .2 o Let p, y be sites in S . { y } δ { p } ⇒ { y } δ V p .3 o V p is close to Vorono¨ı region V y if and only if d ( x, V y ) = 0 for at least one x ∈ V p .4 o A mapping f : V p −→ V y is uniformly continuous, provided f ( V p ) δ f ( V y ) whenever V p δ V y .Proof. o : Assume S contains at least 2 sites. Let p ∈ S, y ∈ S \ { y } such that V p , V y haveat least one closed half plane in common. Then V p δ V y .2 o : If { y } δ { p } , then k y − p k = 0, since y ∈ { y } ∩ { p } . Consequently, { y } ∩ cl( V p ) = ∅ . Hence, { y } δ cl( V p ).3 o : V p δ V y ⇔ exists x ∈ cl( V p ) ∩ cl( V y ) ⇔ d ( x, V y ) = 0. J.F. PETERS o : Let f ( V p ) δ f ( V y ) whenever V p δ V y . Then, by definition, f : V p −→ V y isuniformly continuous. (cid:3) Theorem 3.6.
Every collection of proximal Vorono¨ı regions has a Leader uniformtopology (application of [9] ).Proof.
Assume D has more than one Vorono¨ı region. For each V p ∈ D , find all V y ∈ D that are close to V p . For each V p , this procedure determines a family ofVorono¨ı regions that are near V p . Let τ be a collection of families of proximalVorono¨ı regions. Let A, B ∈ τ . A ∩ B ∈ τ , since either A ∩ B = ∅ or, fromTheorem 3.5.l o , there is at least one Vorono¨ı region V p ∈ A ∩ B , i.e. , V p δ A and V p δ B . Hence, A ∩ B ∈ τ . Similarly, A ∪ B ∈ τ , since V p δ A or V p δ B for each V p ∈ A ∪ B . Also, D , ∅ are in τ . Then, τ is a Leader uniform topology in D . (cid:3) References [1] G. Beer,
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