  We define a notion of extrinsic black hole in pure Lovelock gravity of degree
$k$ which captures the essential features of the so-called
Lovelock-Schwarzschild solutions, viewed as rotationally invariant
hypersurfaces with null $2k$-mean curvature in Euclidean space $\mathbb
R^{n+1}$, $2\leq 2k\leq n-1$. We then combine a regularity argument with a
rigidity result by Ara\'ujo-Leite to prove, under a natural ellipticity
condition, a global uniqueness theorem for this class of black holes. As a
consequence we obtain, in the context of the corresponding Penrose inequality
for graphs established by Ge-Wang-Wu, a local rigidity result for the
Lovelock-Schwarzschild solutions.
