Lipschitzian mappings and total mean curvature of polyhedral surfaces. I

For a smooth closed surface $C$ in ${E^3}$ the classical total mean curvature is defined by $M(C) = \frac{1} {2}\int ({\kappa _1} + {\kappa _2})\;d\sigma (p)$, where ${\kappa _1},{\kappa _2}$ are the principal curvatures at $p$ on $C$. If $C$ is a polyhedral surface, there is a well known discrete version given by $M(C) = \frac{1} {2}\Sigma {l_i}(\pi - {\alpha _i})$, where ${l_i}$ represents edge length and ${\alpha _i}$ the corresponding dihedral angle along the edge. In this article formulas involving differentials of total mean curvature (closely related to the differential formula of L. Schláfli) are applied to several questions concerning Lipschitizian mappings of polyhedral surfaces.

For example, the simplest formula $\Sigma {l_i}\,d{\alpha _i} = 0$ may be used to show that the remarkable flexible polyhedral spheres of R. Connelly must flex with constant total mean curvature. Related differential formulas are instrumental in showing that if $f: {E^2} \to {E^2}$ is a distance-increasing function and $K \subset {E^2}$, then $\operatorname{Per}(\operatorname{conv}\;K) \leqslant \operatorname{Per}(\operatorname{conv}\;f[K])$.

This article (part I) is mainly concerned with problems in ${E^n}$. In the sequel (part II) related questions in ${S^n}$ and ${H^n}$, as well as ${E^n}$, will be considered.