Euclidean ball

This is a ball for the Euclidean metric.

The closed Euclidean ball of radius $R$ about $a\in {\mathbb{R}}^n$ is $\{x:\Vert x-a{\Vert }_2\le R\}$; replacing $\le$ by $<$ gives the open ball.

The Euclidean unit ball is the closed unit ball of the Euclidean norm, centered at zero. Its radial function is identically $1$, making it a natural reference body for John ellipsoids and volume ratios.

The unit ball of a normed vector space is $\{x:\Vert x\Vert \le 1\}$ under the closed-ball convention.

The closed unit ball of a normed vector space is the set of vectors of norm at most one. In a continuous dual space it is compact in the weak-star topology by the Banach-Alaoglu theorem.

The open unit ball of a normed vector space $X$ is $B_X=\{x\in X:\Vert x\Vert <1\}$.