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Irreducible convex sets

Part of: Polytopes and polyhedra

Published online by Cambridge University Press:  26 February 2010

David Yost
David Yost
Affiliation:
Dr. D. Yost, Mathematisches Institut, Freie Universitat Berlin, Arnimallee 3, 1000 Berlin 33, Germany.
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Extract

This paper is concerned with convex subsets of finite dimensional vtctor spaces, over the field of real scalars. As in [10, p. 244] and [20] we say that a compact convex set A, symmetric about the origin, is reducible, if there is a nonsymmetric closed convex set B for which A = B - B. The latter term denotes the set of all differences, {x-y: x, y є B}. Equivalently, A is reducible if, and only if, A =1/2 (B - B) for some B which is not a translate of A. If the identity A = B - B is only possible when B is centrally symmetric, then A is irreducible. If A is symmetric about a point other than the origin, we can say that it is (ir)reducible when it is the translate of an (ir)reducible set which is symmetric about the origin. It is well known that a parallelotope of any dimension is irreducible [6, Hilfssatz 3], that any 2-dimensional convex body other than a parallelogram is reducible [9, p. 217], and that euclidean balls of any dimension (other than one) are reducible [4, Ch. 7]. For more general convex bodies, the determination of reducibility is not a simple problem. Shephard [20] showed that a set is reducible if, and only if, it has an asymmetric summand, and he used this to study reducibility of polytopes. The main purpose of this paper is to give a new condition, necessary and sufficient for a symmetric polytope to be reducible. This condition may be expressed in the form: does a certain finite family of linear equations have a nontrivial solution? Thus, to determine the reducibility of a given polytope, it suffices to find the rank of some matrix. Using our criterion, we are able to describe some large families of irreducible polytopes. For example, every n- dimensional symmetric polytope with 4n – 2 or fewer vertices is irreducible (unless n = 2). We also establish the existence of irreducible, smooth, strictly convex bodies.

MSC classification

Secondary: 52B11: $n$-dimensional polytopes
Type
Research Article
Information
Mathematika , Volume 38 , Issue 1 , June 1991 , pp. 134 - 155
Copyright
Copyright © University College London 1991

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References

1 Balinski, M. L.. On the graph structure of convex polyhedra in n-space. Pacific J. Math., 11 (1961), 431–434.CrossRefGoogle Scholar
2 Bouwkamp, C. J.. On the dissection of rectangles into squares (Second communication). Indag. Math., 9 (1947), 43–56.Google Scholar
3 Bronsted, A. and Maxwell, G.. A new proof of the d-connectedness of d-polytopes. Canad. Math. Bull., 32 (1989), 252–254.CrossRefGoogle Scholar
4 Eggleston, H. G.. Convexity (Cambridge University Press, 1958).CrossRefGoogle Scholar
5 Eggleston, H. G.. Sets of constant width in finite dimensional Banach spaces. Israel J. Math., 3 (1965), 163–172.CrossRefGoogle Scholar
6 Groemer, H.. Abschàtzungen für die Anzahl der konvexen Körper, die einen konvexen Körper berühren. Monatsh. Math., 65 (1961), 74–81.CrossRefGoogle Scholar
7 Gruber, P. M.. Die meisten konvexen Korper sind glatt, aber nicht zu glatt. Math. Ann., 229 (1977), 259–266.CrossRefGoogle Scholar
8 Gruber, P. M.. Results of Baire category type in convexity. Ann. New York Acad. Sci., 440 (1985), 163–169.CrossRefGoogle Scholar
9 Grünbaum, B.. On a conjecture of H. Hadwiger. Pacific J. Math., 11 (1961), 215–219.CrossRefGoogle Scholar
10 Grünbaum, B.. Measures of symmetry for convex sets. Proc. Sympos. Pure Math., 7 (1963), 233–270.CrossRefGoogle Scholar
11 Kirkman, T. P.. Applications of the theory of the polyedra to the enumeration and registration of results. Proc. Roy. Soc. London, 12 (1863), 341–380.Google Scholar
12 Klee, V.. Some new results on smoothness and rotundity in normed linear spaces. Math. Ann., 139 (1959), 51–63.CrossRefGoogle Scholar
13 Lima, Å.. On M-ideals and best approximation. Indiana Univ. Math. J., 31 (1982), 27–36.CrossRefGoogle Scholar
14 Lima, Å. and Yost, D.. Absolutely Chebyshev subspaces. Proc. Centre Math. Anal. Austral. Nat. Univ., 20 (1988), 116–127.Google Scholar
15 McMullen, P.. Indecomposable convex polytopes. Israel J. Math., 58 (1987), 321–323.CrossRefGoogle Scholar
16 Meyer, W.. Indecomposable polytopes. Trans. Amer. Math. Soc, 190 (1974), 77–86.CrossRefGoogle Scholar
17 Payà, R. and Yost, D.. The two-ball property: transitivity and examples. Mathematika, 35 (1988), 190–197.CrossRefGoogle Scholar
18 Reisner, S.. Certain Banach spaces associated with graphs and CL-spaces with 1-unconditional bases. To appear in J. London Math. Soc.Google Scholar
19 Shephard, G. C.. Decomposable convex polyhedra. Mathematika, 10 (1963), 89–95.CrossRefGoogle Scholar
20 Shephard, G. C.. Reducible convex sets. Mathematika, 13 (1966), 49–50.CrossRefGoogle Scholar
21 Smilansky, Z.. An indecomposable polytope all of whose facets are decomposable. Mathematika, 33 (1986), 192–196.CrossRefGoogle Scholar
22 Smilansky, Z.. Decomposability of polytopes and polyhedra. Geom. Ded., 24 (1987), 29–49.CrossRefGoogle Scholar
23 Zamfirescu, T.. Using Baire categories in geometry. Rend. Sem. Mat. Univ. Politec. Torino, 43 (1985), 67–88.Google Scholar