The Mobius-Pompeiu metric property

In the paper we consider an extension of Mobius-Pompeiu theorem of the elementary geometry over metric spaces. We specially take into consideration Ptolemaic metric spaces.


Ptolemaic metric spaces
A metric space (X, d) is called Ptolemaic metric space if Ptolemaic inequality holds: for every x 1 , x 2 , x 3 , x 4 ∈ X [3]. A normed space (X, | . |) is Ptolemaic normed space if metric space (X, d) is Ptolemaic with the distance d(x, y) = |x − y|. Let us notice that the following lemma is true [3]: Lemma 2.1 A normed space is Ptolemaic iff it is an inner product space.
We give two basic examples of Ptolemaic spaces [3]. 2 is a Ptolemaic metric space.
We will illustrate following considerations with the previous examples of Ptolemaic metric spaces in the case of dimension n = 2.

The main results
Let (X, d) be a metric space. Let us fix three points A, B, C ∈ X and form distances: For any point M ∈ X let us form distances: Let us determine a set of M points of metric spaces X for which the following inequality is true: Let us form two functions: Proof. For point A: d 1 = 0 and α 1 = −(c 2 − b 2 ) 2 ≤ 0 are true. Similarly, the previous inequality is true for the points B and C.
Example 3.2 Let vertices A, B, C of the triangle ABC in the plane R 2 be given by coordinates A(a 1 , b 1 ), B(a 2 , b 2 ), C(a 3 , b 3 ) and let M (x, y) be any point in its plane.
1 0 . Let us in the plane R 2 use Euclidean metric d. Let us specify the form of term α 1 and β 1 which correspond to functions (3.4) and (3.5) respectively. It is true: . Equality α 1 = 0 determines the algebraic curve of the fourth order. By inequality α 1 < 0 we determine the interior of the previous curve. Also, it is true: for some coefficients A 2 , B 2 , C 2 , D 2 ∈ R (A 2 = 1). If B 2 2 + C 2 2 > 4D 2 equality β 1 = 0 is possible and determines the circle. Then by inequality β 1 < 0 we determine the interior of the circle.
for some coefficients k, A 1 , B 1 , C 1 , D 1 , E 1 , F 1 , G 1 , H 1 ∈ R. If k = 0 equality α 1 = 0 determines the algebraic curve of the fourth order. Then by inequality α 1 < 0 we determine the interior of the previous curve. Also, it is true: is possible and determines the circle. Then by the inequality β 1 < 0 we determine the interior of the circle.
Further, let us notice that for the function α 1 : According to (3.10) equality α 1 = 0 is equivalent with union of equalities: Subject to our further consideration is an inequality α (1) where equality is true for M = B and a = c (3.14) and Hence, the inequality (3.14) follows. Thus, the equality is true only if M = B (d 2 = 0) and a = c. Analogously, it is true Hence, the inequality (3.15) follows. Thus, the equality is true only if M = C (d 3 = 0) and a = b. Lemma 3.6 1 0 . If the point M fulfills d 2 + d 3 ≤ d 1 then the following implication is true: Proof. The implications (3.18) and (3.19) have the same assumptions: which follow if the following conjunction is true or the conjunction 18) is directly verified. Especially for M = B and a = c or for M = C and a = b equality β 1 = 0 is true. Let us assume that M = B, C and let us assume that α 1 ≤ 0 in (3.18) be true. On the basis of d 2 + d 3 ≤ d 1 , according to lemma 3.5 it follows that Especially for M = B and a = c or for M = C and a = b equality β 1 = 0 is true. Let us assume that M = B, C and let us assume that α 1 ≤ 0 in (3.19) be true. On the basis of d 3 + d 1 ≤ d 2 , according to the lemma analogous to lemma 3.5, it follows d 2 + d 3 > d 1 and d 1 + d 2 > d 3 . Therefore Lemma 3.7 In the metric space X the condition d 2 + d 3 ≤ d 1 is equivalent to the conjunction α 1 ≤ 0 and β 1 ≤ 0.
Proof. (=⇒) Let for the point M the condition d 2 + d 3 ≤ d 1 be true. On the basis of equality (3.10) and on the basis of lemma 3.5 it follows α 1 ≤ 0. Therefore, on the basis of lemma 3.6, it follows β 1 ≤ 0.
(⇐=) Let for the point M conjunction α 1 ≤ 0 and β 1 ≤ 0 be true. Then from the conjunction Lemma 3.8 In Ptolemaic metric space X an inequality α (1) and assumption a > b, c we can conclude By contraposition the statement follows.
On the basis of the previous lemmas we can conclude the following theorem is true.
Let us determine set of points M in (Ptolemaic) metric spaces for which some inequalities in (1.3) are true. With respect to point A we formed functions (3.4) and (3.5). Next, with respect to point B let us form functions: and with respect to C point let us form functions: The following equality α 1 = α 2 = α 3 is true. Analogously to the theorem 3.9 we can conclude the following theorems are true. Using theorems 3.9, 3.10 and 3.11 we can determine when some inequalities in (3.37) are not true.
Finally, in the following example let us illustrate a set of points in R 2 with Möbius-Pompeïu metric property, with respect to three fixed points A, B, C ∈ R 2 , if we use metrics d and d from the example 2.2.
1 = 0 have non-empty interior and border. We can form a non-degenerative triangle from the remaining points.
In the case of the equilateral triangle ABC the curves α 2 0 . Let in the plane R 2 the chordal metric d is used. Let A, B, C ∈ S\{(0, 0, 1)} be points on the unit Riemann sphere S, with uniquely determined projections: with inversely stereographical projection from the north pole: Through points A, B, C on the Riemann sphere let us set great circles (picture 2). In the complex plane we uniquely determine images of great circles as corresponding circles through points A ′ , B ′ , C ′ (picture 3). The Möbius-Pompeïu metric property