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$h = distance(z, \tilde{z})$, where $\tilde{z}$ is the element that is nearest from $z$ (that is, distance(z, $\tilde{z}$) is smaller than distance(z, any_other_z)).

Is it possible to expression this formally, instead of saying "where $\tilde{z}$ is the element that is nearest from $z$ ..." ?

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up vote 1 down vote accepted

There are two possible answers to your question.

  1. If you just want an expression for $\tilde{z}$, you can use $\operatorname{argmin}$, like this: $h = \operatorname{distance}(z,\operatorname{argmin}_{\tilde{z} \neq z} \operatorname{distance}(z,\tilde{z}))$.

Formally, $\operatorname{argmin}_{x \in S} f(x)$ is defined as any value $x \in S$ such that $f(x)$ is minimal.

  1. Note that your expression is identical to $$\min \{ \operatorname{distance}(z,\tilde{z}) | z \ne \tilde{z} \}.$$ This is probably the best solution.
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Great. By the way, is it possible that $z \neq \tilde{z}$ is under argmin in latex (and not just an index) ? – shn Jul 8 '12 at 15:33
Yes (works directly when you use display-style math), or by doing some tricks with limits/nolimits. – Johannes Kloos Jul 8 '12 at 17:32

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