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In my book, I came across this set:

$$\{\mathbf{x}\in\mathbb{R}^n:\|\mathbf{x}\|=1\}$$

where $\|\mathbf{x}\|$ is the Euclidean norm.

Is there a name for this kind of set?

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    $\begingroup$ It is the $(n-1)$-sphere, typically denoted by $S^{n-1}$. See here $\endgroup$ – Hayden Apr 16 '17 at 5:41
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    $\begingroup$ @Hayden, thanks! You can provide an answer, so that I may select it (for the sake of having an answer to the question). $\endgroup$ – Jake Apr 16 '17 at 5:44
  • $\begingroup$ It's called a sphere. $\qquad$ $\endgroup$ – Michael Hardy Apr 16 '17 at 6:01
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This is called the unit sphere in $\mathbb{R}^n$. It utilizes the $\ell_2$ norm, which is the standard Euclidean norm. The unit sphere for other norms will look different, for example using the $\ell_1$ norm in $\mathbb{R}^2$ gives you a diamond shape, and the $\ell_\infty$ norm gives a square.

The (closed) unit ball is a similar set, $\{ x \in \mathbb{R}^n : \|x\| \leq 1 \}$, which is a unit sphere plus all the points contained within it.

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This is called a unit sphere. In the case $n=1$, it contains just two points; in the case $n=2$ it's the unit circle; in the case $n=3$ it's the sphere that you called a sphere in high-school geometry.

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