  We express the $q$-th Gauss--Bonnet--Chern mass of an immersed submanifold of
Euclidean space as a linear combination of two terms: the total $(2q)$-th mean
curvature and the integral, over the entire manifold, of the inner product
between the $(2q+1)$-th mean curvature vector and the position vector of the
immersion. As a consequence, we obtain, for each $q$, a geometric inequality
that holds whenever the positive mass theorem (for the $q$-th
Gauss--Bonnet--Chern mass) holds.
