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?

  • 1
    $\begingroup$ What do you mean by a distance which is not metric? $\endgroup$ – Moishe Kohan Jul 10 at 3:17
  • $\begingroup$ 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. $\endgroup$ – 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 $1$.

  • $\begingroup$ This is definitely a perfectly valid answer, thank you! I hope you don't mind that I'll stake out a little bit longer in the hopes of a "real" example before accepting your answer. :) $\endgroup$ – Kjeld Schmidt Jul 10 at 9:14
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    $\begingroup$ @KjeldSchmidt No problem. I agree with you that this doesn't really feel like a ""real" example. $\endgroup$ – Jeremy Rickard Jul 10 at 10:12

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