A NON-UNIQUENESS THEOREM IN THE THEORY OF VORONOI SETS

It is shown that two distinct, bounded, open subsets of ℝ2 may possess the same Voronoi set.


INTRODUCTION
Let {Di}o<i_<n be a finite collection of non-empty, bounded,  For any (x,y) , define Near(x,y) as the set of points in closest to (x,y).
("Closest to" is, of course, defi,ed in terms of ordinary Euclidean distance in the plane.)Since is closed, Near(x,y) is always non-empty.
The Vul% d Vor() of is then defined to be the set of points {(x,y) Near(x,y) contains more than one point}.
Vor() is used in [I] in connection with motion planning problems.
Clearly given the sets {Di} Vor() is unique.However, here we take the opposite point of view and consider the construction of the sets {D.} from a given I Voronoi diagram.
A preliminary question that one might ask is: could it be possible for two collections {D.} and {D} to have the same Voronoi diagrams?It is easy to see that I the answer is yes: for O < e I let DOe {(x,y) x+y < (l+e) 2} and 2 2 2 DI ((x,y) x +y (i-) }.

H. JONES
Then if e DO\I, Vor(e) is the unit circle, centre the origin, whatever the value of E night be.
A more subtle question is the following: Suppose D O D, then is it possible for two different collections {D.} and {D} to have the same Voronoi diagram?
Informally, what we are asking is whether, given a fixed domain DO, it is possible to arrange two different sets of obstacles within DO, both of which produce tile same Voronoi diagram.) We show the answer is again in the affirmative.--/ (4,4) ")"-3,3) To see that the Voronoi diagrams of and ' are indeed the same first note that it suffices to consider those points (x,y) in ' for which Ixl -< I and IYl < /2 since for any other (x,y) E ', Near(x,y) will be unchanged by the modifications made to D I and D 2. To begin with, consider those points the triangle whose vertices are (-i,0), (O,O) and (-i,i).It is clear that if (x,y) is such a point then Near(x,y) {(-i,I)} and so (x,y) 4 '.The same conclusion is true for the points in ' which lie on the straight lines joining (-I,I) to (-i,0)   and (-i,I) to (O,O), (excluding the endpoints of those lines).Next consider the points (x,O) where -i < x < O.For such a point Near(x,O) {(-i,i), (-I,-I)} and so (x,O) E Vor(').It is also clear that (O,O)   Vor(').Now consider those points within the sector of C which has vertices (0,0), (-i,I) and (0,/2).If (x,y) is such a point then it is easy to see that Near(x,y) consists of the single point obtained by projecting the straight line joining (O,0) to (x,y) until it

THE EXAMPLE
The same conclusion is true for the points on the straight line intersects D I.
between (0,O) and (0,/2) (excluding the endpoints of course).The results for containing "obstacles" Di, i E i n.)The the following definition of the Voronoi diagram Vor() of is taken from [I].
Then and Vor(.) (where Do\D I u D2) are depicted in Figure i.Note in particular that Vor([) contains the line segment {(x,O)[ Ix < I}.

Figure 2
Figure IVor() is denoted by the dashed line