It should be understood as "For any $R>0$, $\inf_{|x_i|<R}\varepsilon_i > 0$".

Quermassintegrals can be defined for arbitrary smooth open subsets of $R^d$ (actually only $C^2$-regularity is necessary). More precisely, they can be expressed as integrals of appropriate functions of the (extrinsic) curvature tensor of the boundary. See, for example, the introduction of this paper by Trudinger (formula (1.3)).