Ball

$B( a, r ) = \{ x \in {\bf R}^n: | x - a | \leq r \}$, a closed ball, $B( a, r ) = \{ x \in {\bf R}^n: | x - a | < r \}$, an open ball.

Which one should be clear from the context.

The boundary of the $n$-ball is an $(n-1)$-sphere.