The Strange World of the Hausdorff Metric Geometry

 

V. The Geometry of the Hausdorff Metric:Circles

 

To build a geometry, we need a collection of points along with lines that connect the points. In the hyperspace H(Rn), the points are elements of H(Rn), namely the non-empty compact subsets of Rn. So the geometry we will study is one in which the points are actual physical objects, not the dimensionless points of Euclidean geometry. To build a geometry, we need to define lines to connect these elements. In the GVSU - REU 2000, Dominic Braun (then at the University of North Carolina - Asheville, now at the University of Virginia) and I extended the standard Euclidean notions of circles and lines to the space H(Rn). We will begin with circles. A word on terminology: to distinguish between “points” in the spaces Rn and H(Rn), we will use the word “point” to refer to a point in Rn and the word “element” when referring to a point in H(Rn).

 

Since we define a circle in Euclidean space to be the set of points equidistant from a given point, why not do the same in H(Rn)? A circle in H(Rn) centered at an element B in H(Rn) with radius r > 0 will be the set of elements A in H(Rn) so that h(B, A) = r. One of the results of the 2000 REU was that Dominic and I developed a complete characterization of circles in H(Rn), as described in the following theorem (Theorem 2 from [7]) which can be formulated as follows:

 

Theorem 1 (The Circle Theorem): Let A be an element of H(Rn) and let r > 0. An element B in H(Rn) satisfies h(B, A) = r if and only if

1.         B is a subset of  A+r,

2.         For each a in A, B intersects the closed ball of radius r centered at a, and

3.         At least one of the following is satisfied:

            (a) B ∩ ∂Nr(A) ≠ Ø, or

            (b) there is a point a in A such that B ∩ ∂Nr(a) ≠ Ø and B ∩ Nr(a) = Ø.

 

To illustrate the Circle Theorem, let A be a two-point set {a1, a2} (in red in the next applet) and B the blue disk. Parts (3a) and (3b) of the Circle Theorem show that B must contain a point on the boundary of one of Nr(a1) or Nr(a2) (drawn in red). This necessary point appears in green. The user can move the green point along boundaries of Nr(a1) and Nr(a2), change the center of B, the location of the points a1 or a2, or the value of r (via the black point). The program indicates the state (true/false) of each of the conditions of the Circled Theorem for the given configuration. Play around to see how B must be situated to satisfy conditions 1, 2, and one of 3(a) or 3(b) and lie on the Hausdorff Circle centered at A.

 

 

The characterization in the Circle Theorem turns out to be very useful in proving facts about Hausdorff lines as well as other properties of the space H(Rn) [14].

 

Dominic and I also began studying lines in H(Rn).  The next applet provides an illustration of the different circles centered at the red rectangle A. Moving the black point changes the radius, r, of the circle. Dominic and I proved that every element in H(Rn) that is r units from A must be a subset of A+r. So while there are many elements on the  circle centered at A of radius r, when drawn all together, they form A+r.