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Assume $r > 0$. Is it standard to use the expression "the $\ell_2$-norm ball of radius $r$" to refer to the set

\begin{equation} B = \{ x \in \mathbb R^n \mid \| x \|_2 \leq r \}. \end{equation}

(Do I also need to state where the ball is centered, or is an "$\ell_2$-norm ball" by definition centered at the origin.)

What's the shortest/easiest/most clear way to refer to $B$?

eternalGoldenBraid
  • 2,089
  • It's acceptable, although if there is no confusion possible, you can simply refer to it as the ball of radius $r$ (without specific reference to the norm).

    However, balls are usually open, so when you refer to a closed one, it's better to talk about the closed ball of radius $r$. If you say 'the' ball, then most people understand that it is centered at the origin, whereas a ball of radius $r$ can be centered anywhere.

     – Mathematician 42 May 01 '16 at 10:00
  • Couldn't there also be some possible confusion as to w.r.t. what space we define the norm on? Sure, $\mathbb{R}^n$ is the typical space that jumps to mind, so the risk is a bit low, but there are others out there too… – MonadBoy May 01 '16 at 10:03
  • As it's centred at $0$, you could call it the closed $r$-ball in $\ell_2$. – Henno Brandsma May 01 '16 at 10:17
  • Euclidean ball of radius $r$, for instance. – Siminore May 01 '16 at 11:33

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