Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

Lipschitzian mappings and total mean curvature of polyhedral surfaces. I


Author: Ralph Alexander
Journal: Trans. Amer. Math. Soc. 288 (1985), 661-678
MSC: Primary 52A25; Secondary 53C45
MathSciNet review: 776397
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: 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.


References [Enhancements On Off] (What's this?)

  • [1] Ralph Alexander, The circumdisk and its relation to a theorem of Kirszbraun and Valentine, Math. Mag. 57 (1984), no. 3, 165–169. MR 740325, 10.2307/2689665
  • [2] R. V. Ambartzumian, On some topological invariants in integral geometry, Z. Wahrsch. Verw. Gebiete 44 (1978), no. 1, 57–69. MR 0488212
  • [3] Wilhelm Blaschke, Vorlesungen über Integralgeometrie, Deutscher Verlag der Wissenschaften, Berlin, 1955 (German). 3te Aufl. MR 0076373
  • [4] J. Böhm and E. Hertel, Polyedergeometrie in $ n$-dimensionalen Räumen konstanter Krümmung, Birkhäuser Verlag, Basel, 1981.
  • [5] G. D. Chakerian, Integral geometry in the Minkowski plane, Duke Math. J. 29 (1962), 375–381. MR 0188961
  • [6] Robert Connelly, A counterexample to the rigidity conjecture for polyhedra, Inst. Hautes Études Sci. Publ. Math. 47 (1977), 333–338. MR 0488071
  • [7] William J. Firey, Christoffel’s problem for general convex bodies, Mathematika 15 (1968), 7–21. MR 0230259
  • [8] Branko Grünbaum, Convex polytopes, With the cooperation of Victor Klee, M. A. Perles and G. C. Shephard. Pure and Applied Mathematics, Vol. 16, Interscience Publishers John Wiley & Sons, Inc., New York, 1967. MR 0226496
  • [9] M. Kirszbraun, Über die zusammenziehenden und Lipschitzschen Transformationen, Fund. Math. 22 (1934), 77-108.
  • [10] Victor Klee, Some unsolved problems in plane geometry, Math. Mag. 52 (1979), no. 3, 131–145. MR 533432, 10.2307/2690274
  • [11] Luis A. Santaló, Integral geometry and geometric probability, Addison-Wesley Publishing Co., Reading, Mass.-London-Amsterdam, 1976. With a foreword by Mark Kac; Encyclopedia of Mathematics and its Applications, Vol. 1. MR 0433364
  • [12] M. Spivak, A comprehensive introduction to differential geometry, Publish or Perish, Berkeley, Calif., 1979.

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 52A25, 53C45

Retrieve articles in all journals with MSC: 52A25, 53C45


Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1985-0776397-6
Article copyright: © Copyright 1985 American Mathematical Society