In this talk we explain how every even, zonal measure on the Euclidean unit sphere gives rise to a sharp Sobolev inequality for functions of bounded variation which directly implies the classical Euclidean Sobolev inequality. The strongest member of this large family of inequalities is shown to be the only affine invariant one among them — the affine Zhang–Sobolev inequality. We also relate our new Sobolev inequalities to the sharp Gromov–Gagliardo–Nirenberg Sobolev inequality for general norms and discuss further improvements of special cases.