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)



Indecomposable polytopes

Author: Walter Meyer
Journal: Trans. Amer. Math. Soc. 190 (1974), 77-86
MSC: Primary 52A25
MathSciNet review: 0338929
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: The space of summands (with respect to vector addition) of a convex polytope in n dimensions is studied. This space is shown to be isomorphic to a convex pointed cone in Euclidean space. The extreme rays of this cone correspond to similarity classes of indecomposable polytopes. The decomposition of a polytope is described and a bound is given for the number of indecomposable summands needed. A means of determining indecomposability from the equations of the bounding hyperplanes is given.

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

  • [1] J. D. Flowers, Facial cones, local similarity and indecomposability of polytopes, Ph D. Thesis, Oklahoma State University, Stillwater, Okla., 1966.
  • [2] D. Gale, Irreducible convex sets, Proc. I.C.M. Amsterdam, 1954, II, 217-218.
  • [3] B. Grünbaum, Measures of symmetry for convex sets, Convexity, Proc. Sympos. J Pure Math., vol 7, Amer. Math. Soc., Providence, R. I., 1963, pp. 233-270. MR 27 #6187. MR 0156259 (27:6187)
  • [4] -, Convex polytopes, Pure and Appl. Math., vol. 16, Interscience, New York, 1967. MR 37 #2085.
  • [5] P. McMullen, Representations of polytopes and polyhedral sets (to appear). MR 0326574 (48:4917)
  • [6] W. Meyer, Minkowski addition of convex sets, Ph D. Dissertation, University of Wisconsin, Madison, 1969.
  • [7] G. C. Shephard, Decomposable convex polyhedra, Mathematika 10 (1963), 89-95. MR 30 #2398. MR 0172176 (30:2398)
  • [8] -, The Steiner point of a convex polytope, Canad. J. Math. 18 (1966), 1294-1300. MR 35 #4814. MR 0213962 (35:4814)

Similar Articles

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

Retrieve articles in all journals with MSC: 52A25

Additional Information

Keywords: Convex cone, decomposition, extreme ray, indecomposable, polytope, vector addition, Minkowski addition
Article copyright: © Copyright 1974 American Mathematical Society

American Mathematical Society