The Strange World of the Hausdorff Metric Geometry


VI. The Geometry of the Hausdorff Metric: Lines


To mimic the properties of Euclidean lines, we are interested in studying sets that satisfy one of the three triangle equalities in the space H(Rn). More specifically, given distinct elements A and B in H(Rn), we are interested in elements C in H(Rn) which satisfy one of the following equalities:


I.          h(A, B) = h(A, C) + h(C, B)

II.         h(A, C) = h(A, B) + h(B, C)    

III.       h(C, B) = h(C, A) + h(A, B)


We will say that sets C satisfying I lie “between” A and B, sets satisfying II lie to the “right” of B, and any set satisfying III is said to lie to the “left” of A. We will use the notation ACB to indicate that C lies between A and B. We will denote the collection of elements C in H(Rn) satisfying any of the equalities I, II, or III by L(A, B) and call this collection the Hausdorff line defined by A and B. Similarly, we will let l(a, b) denote the Euclidean line through the points a and b in Rn.


For example, let A, B, and C be concentric circles as shown below. You can vary the radii by moving the points. Notice that the relative positions of the elements A, B, and C change as expected as we alter radii. Note also that the circles C are not necessarily the only elements in H(Rn) satisfying the equalities I, II, or III for the given A and B.



For the moment, we will say that a line in H(Rn) defined by two distinct elements A and B in H(Rn) is the set of all elements C of H(Rn) satisfying one of the equalities I, II, or III. Later we will see that this definition is not totally satisfactory.