# General setting for triangle inequalities (terminology question)

Regarding the "mathematical object $(X, s)$" described below (in general, or under some more specific conditions) I'd like to know whether it is called a particular name, for reference in the literature:

How do you call a set $X$ together with a map $s\!\!: X^2 \rightarrow R$

(0) in general, or

(1) under the condition that $\forall \eta \in X: s[ \, \eta, \eta \, ] = 0$, or

permitting that there may $\exists \eta, \phi \in X: s[ \, \eta, \phi \, ] \lt 0$, under the additional condition that:

(2) for any three elements $\rho$, $\sigma$ and $\tau \in X$
$(0 \geq s[ \, \rho, \sigma \, ] \geq s[ \, \sigma, \tau \, ] \geq s[ \, \tau, \rho \, ]) \implies$ $(s[ \, \rho, \sigma \, ] + s[ \, \sigma, \tau \, ] \geq s[ \, \tau, \rho \, ])$

?

A remark on ínequalities involving values $s[ \, \alpha, \beta \, ] \gt 0$ for suitable elements $\alpha$, and $\beta \in X$:

Considering 7 suitable elements of $X$, with corresponding 21 values
$s[ \, \omega, \kappa \, ] = 0$, $s[ \, \omega, \lambda \, ] = 0$, $s[ \, \omega, \mu \, ] = 0$, $s[ \, \omega, \nu \, ] = 0$, $s[ \, \mu, \rho \, ] = 0$, $s[ \, \nu, \sigma \, ] = 0$, and

$s[ \, \kappa, \lambda \, ] \lt 0$, $s[ \, \mu, \nu \, ] \lt 0$,
$s[ \, \kappa, \nu \, ] \leq 0$, $s[ \, \lambda, \mu \, ] \leq 0$,
$s[ \, \kappa, \mu \, ] \gt 0$, $s[ \, \lambda, \nu \, ] \gt 0$; and also

$s[ \, \rho, \sigma \, ] \lt 0$, $s[ \, \omega, \rho \, ] \leq 0$, $s[ \, \omega, \sigma \, ] \leq 0$, and moreover

$0 \geq s[ \, \kappa, \rho \, ] \geq s[ \, \omega, \rho \, ]$,
$0 \geq s[ \, \lambda, \sigma \, ] \geq s[ \, \omega, \sigma \, ]$, and finally

$s[ \, \kappa, \sigma \, ] \lt 0$,
$s[ \, \lambda, \rho \, ] \lt 0$,
$s[ \, \mu, \sigma \, ] \lt s[ \, \mu, \nu \, ] \lt 0$, and
$s[ \, \nu, \rho \, ] \lt s[ \, \mu, \nu \, ] \lt 0$,

then ("by construction") holds:

$s[ \, \rho, \sigma \, ] \leq s[ \, \kappa, \lambda \, ] \lt 0$.

Consequently, in case there exist suitable elements $\alpha$, and $\beta \in X$ for which

$s[ \, \kappa, \alpha \, ] \lt 0$, $s[ \, \alpha, \lambda \, ] \lt 0$,
$s[ \, \mu, \beta \, ] \lt 0$, $s[ \, \beta, \nu \, ] \lt 0$, and

$s[ \, \alpha, \beta \, ] \gt 0$, $s[ \, \alpha, \omega \, ] \gt 0$, $s[ \, \beta, \omega \, ] \gt 0$ such that

$\frac{s[ \, \alpha, \omega \, ]}{s[ \, \kappa, \lambda \, ]} = \frac{s[ \, \beta, \omega \, ]}{s[ \, \mu, \nu \, ]} = \frac{s[ \, \alpha, \beta \, ] + s[ \, \beta, \omega \, ]}{s[ \, \rho, \sigma \, ]}$,

then the following triangle inequality holds: $s[ \, \alpha, \omega \, ] \leq s[ \, \alpha, \beta \, ] + s[ \, \beta, \omega \, ]$.

An example for a "space" (in the sense of a mathematical object) satisfying all these conditions may be any set of "events" which are pairwise related being "spacelike", or "lightlike", or "timelike" to each other, such that

• any pair of "spacelike" events are characrerized by positive quasi-distance values among each other, $(s[ \, \alpha, \beta \, ] > 0$, and $s[ \, \beta, \alpha \, ] > 0$,

• any pair of "timelike" events are characrerized by a (proper) duration value among each other, corresponding to $0 \geq s[ \, \rho, \sigma \, ] = s[ \, \sigma, \rho \, ]$,

• for any pair of "spacelike" events there are events "lightlike" wrt. both, and

• for any pair of "timelike" events there are events "lightlike" wrt. both, where

• any pair of "lightlight" events would be assigned the value $s[ \, \tau, \alpha \, ] = 0 = s[ \, \alpha, \tau \, ])$.

(This is an edited, third version of my question. The comments below, of January 2012, were based on a less organized first version. The second version, which had been published meanwhile and on which no further comments were issued, should hereby be considered withdrawn, as flawed.)

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Do you have an example of such an object $(X,s)$? –  Mahdi Majidi-Zolbanin Jan 23 '12 at 21:22
If $s$ is any nonnegative function, then your condition is satisfied automatically. –  Ilya Bogdanov Jan 24 '12 at 7:13