locally nonsatiated preferences what does this symbol mean in the discuss of locally nonsatiated preferences:
$\varepsilon &gt; 0$ and $||y-x||&lt;\varepsilon$.
If you take the vector $\mathbf{x}$ as the center and an $\epsilon &gt; 0$ sufficiently small as the radius, then the open ball (local neighborhood of the vector $\mathbf{x}$) is the set of all vectors $\mathbf{y} \in \mathbb{R}^{n}$ whose distance to the center $\mathbf{x}$ is less than $\epsilon$ measured w.r.t. the metric $\Arrowvert \cdot \Arrowvert $, i.e., more formally as

$$\mathbf{B}(\mathbf{x},\epsilon) := \big\{\mathbf{y} \in \mathbb{R}^{n}\,\Big\arrowvert\, \Arrowvert \mathbf{y} - \mathbf{x} \Arrowvert &lt; \epsilon \big\}.$$

Notice that measuring a distance by a metric is only possible within a metric space or normed space like Hausdorff or Banach. Topological spaces do not have in general a metric, in this case the above concept must be generalized. Fortunately, all finite dimensional spaces have a canonical norm, i.e., the Euclidean norm.