Real Balls

Definition

Let $x \in \R ^d$. The open ball centered at $x$ (or around $x$) of radius $\delta > 0$ is the set

\[ \Set*{y \in \R ^{d}}{d(x, y) < \delta } \]

where $d: \R ^d \times \R ^d \to \R $ is the usual Euclidean distance.

We sometimes denote the open ball by $B(x,\delta )$ or $B_{\delta }(x)$