What does this symbol $\mathcal{B}_{\infty}$ stand for？

Question

What is the meaning of $\mathcal{B}_{\infty}(0,4\pi)$?

Answer 1

Well, depends on context, but the general interpretation is: $B_k(x,r) = \{y \in \mathbb{R}^n\ | \ \|x-y\|_k < r \}$ in this case, we use $\|x\|_\infty$ as the greater value between all components of $x$, for example, if $n=3$ and $x=(1,2,5)$ for example, $\|x\|_\infty = 5$

So $B_\infty(0,4\pi) = \{y \in \mathbb{R}^n \ | \ \|x\|_\infty < 4\pi\} = \{(x_1,\cdots,x_n) \ |\ |x_i| < 4\pi \ \ i = 1,2,\cdots.n\}$ It is read as "Infinity Norm ball centered at $0$ with radius $4\pi$"