The Möbius-pompeïu Metric Property

Let us consider analogous problem for the metric space (X, d) with at least four points. Let A,B,C ∈ X be three fixed points. Then, for the point M ∈ X we suppose that a triangle can be formed from the distances d1 = d(M,A), d2 = d(M,B) and d3 = d(M,C) iff the following conjunction of inequalities is true: d1 + d2 − d3 ≥ 0 and d2 + d3 − d1 ≥ 0 and d3 + d1 − d2 ≥ 0. (1.1) If in conjunction (1.1) at least one equality is true, then we suppose that a degenerative triangle can be formed. If in (1.1) sharp inequalities are true: d1 + d2 − d3 > 0 and d2 + d3 − d1 > 0 and d3 + d1 − d2 > 0, (1.2) then we suppose that a non-degenerative triangle can be formed. In this case, for the point M , for which the conjunction (1.2) is true, we define that point have Möbius-Pompeïu metric property. The main subject of this paper is to determine points M which do not have Möbius-Pompëiu metric property, i.e. these points which fulfill the following disjunction of the inequalities: d1 + d2 − d3 ≤ 0 or d2 + d3 − d1 ≤ 0 or d3 + d1 − d2 ≤ 0. (1.3) Let us notice that the point M ∈ {A,B,C} do not have Möbius-Pompëiu metric property. Thus in consideration which follows, we assume that the metric space (X, d) has at least four points. 2000 Mathematics Subject Classification: 54E35, 51M16. Research partially supported by the MNTRS, Serbia & Montenegro, Grant No. 1861.