In this paper, some sharp Sobolev inequalities on $BV({\mathbb{R}}^n)$, the space of functions of bounded variation on ${\mathbb{R}}^n$, $n\ge 2$, are deduced through the $L_p$ Brunn-Minkowski theory. We will prove that these inequalities can all imply the sharp Sobolev inequality on $BV({\mathbb{R}}^n)$.