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