Neighborhood

In mathematics, a neighborhood of a point p\in V is a subset of V consisting of points that are within a given distance from p. This is illustrated in the picture (from WIKIPEDIA), where the neighborhood of p is a disk of a certain radius, centered at p.

Note that this interpretation of the term neighborhood relies on the existence of some measure of “distance” between points in V. So, more explicitly, we can define a neighborhood as follows:

    \displaystyle \mathcal{N}(\rho,p):= \{q\in V: dist(q,p) \le \rho\}, \rho\ge 0, p\in V

where dist(q,p) denotes the distance from q to p.

We shall not dwell on properties of the distance function dist. For the purposes of this discussion it would suffice to assume that the neighborhoods induced by this function have the following two properties:

  • \mathcal{N}(0,p) = \{p\} , \forall p\in V
  • \rho' < \rho \ \longrightarrow \ \mathcal{N}(\rho',p) \subseteq \mathcal{N}(\rho,p)

In some areas of mathematics, neighborhoods are called balls.

An analysis, e.g. robustness analysis, that is conducted in the neighborhood of a point p\in V is local. In contrast, a global analysis is an analysis that is conducted on the entire given set V. Hence the difference between local minima and global minima.

Usually, the neighborhoods of interest are much smaller than the metropolitan set V that contains them.

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