Strengthening of the triangle inequality

Rok, strany: 1996, 211 - 216

We deal with a certain way of strengthening of the triangle inequality. The main result could be formulated as follows: \proclaim{\small{Theorem}} Let $1≥ a>0$. Let $g:\langle 0,a)\to \Bbb R⁺₀$ be a non-decreasing function such that $g(0)=0$ and $0<g(x)≤ x$ for any $x\in (0, a)$. Let $(X,d)$ be a metric space. Then there exists the metric $d₁$ on $X$ such that topologies on $(X,d)$ and $(X,d₁)$ coincide and the metric $d₁$ satisfies the following “strong” triangle inequality: for $x,y,z\in X$

$$ d(x,y)≤ \max \{d(x,z),d(y,z)\} + g(\min\{d(x,z),d(y,z)\}). $$

ISO 690:

Guričan, J. 1996. Strengthening of the triangle inequality. In Tatra Mountains Mathematical Publications, vol. 8, no.2, pp. 211-216. 1210-3195.

APA:

Guričan, J. (1996). Strengthening of the triangle inequality. Tatra Mountains Mathematical Publications, 8(2), 211-216. 1210-3195.