Is there a name for this set?

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?

• It is the $(n-1)$-sphere, typically denoted by $S^{n-1}$. See here – Hayden Apr 16 '17 at 5:41
• @Hayden, thanks! You can provide an answer, so that I may select it (for the sake of having an answer to the question). – Jake Apr 16 '17 at 5:44
• It's called a sphere. $\qquad$ – Michael Hardy Apr 16 '17 at 6:01

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.
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.