# Is there an example of a distance which is ptolemaic but not metric?

Ptolemaic distances are distances for which the property

$$\overline{x_1x_3} \cdot \overline{x_2x_4} \leq \overline{x_1x_2} \cdot \overline{x_3x_4} + \overline{x_2x_3} \cdot \overline{x_1x_4}$$

holds for any four points $$x_1, x_2, x_3, x_4$$. Examples include the euclidean metric, more generally all metrics induced by an inner product. Metrics which are not ptolemaic include the $$L_p$$-distances for $$p \neq 2$$. In fact, the $$L_p$$-distances are neither metric nor ptolemaic for $$p \in (0, 1)$$.

This leaves but a single category: Distances which are ptolemaic but not metric. Are there any examples of this?

• What do you mean by a distance which is not metric? – Moishe Kohan Jul 10 at 3:17
• A distance which is not a metric fulfills: Being positive semidefinite - Being symmetric - having identity of indiscernables, i.e. the function is 0 iff the two arguments are equal. However, it does not fulfill the triangle equality. For additional formalism, a distance takes two inputs from the same arbitrary domain and returns a non-negative real number. – Kjeld Schmidt Jul 10 at 3:40

Perhaps a rather silly example, but there is a non-metric ptolemaic distance on a set $$\{x_1,x_2,x_3,x_4\}$$ with only four elements.
Let the distance from $$x_1$$ to $$x_2$$ be $$1000000$$, the distance from $$x_3$$ to $$x_4$$ be $$0.000001$$ and all other distances between distinct points be $$1$$.