Convex Polytopes, Dihedral Angles, Mean Curvature and Scalar Curvature

We approximate boundaries of convex polytopes \(X\subset {\mathbb {R}}^n\) $X\subset {\mathbb{R}}^n$ by smooth hypersurfaces \(Y=Y_\varepsilon \) $Y=Y_{\epsilon }$ with positive mean curvatures and, by using basic geometric relations between the scalar curvatures of Riemannian manifolds and the mean curvatures of their boundaries, establish lower bound on the dihedral angles of X.