# 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.