Abstract

Following R. T. Rockafellar ["Convex Analysis", Princeton University Press, Princeton (1970)], a generalized nn-simplex in Rn\R^n is defined as the direct sum of an mm-simplex and a simplicial (n−m)(n - m)-cone, 0≤m≤n0 \le m \le n. R. Fourneau ["Nonclosed simplices and quasi-simplices", Mathematika 24 (1977) 71–85] showed that a line-free nn-dimensional closed convex set K⊂RnK \subset \R^n is a generalized nn-simplex if and only if all nn-dimensional intersections K∩(v+K)K \cap (v + K), v∈Rnv \in \R^n, are homothetic to KK. We extend this characteristic property by proving that for a pair of line-free nn-dimensional closed convex sets K1K_1 and K2K_2 in Rn\R^n the following two conditions are equivalent: (1) all nn-dimensional intersections K1∩(v+K2)K_1 \cap (v + K_2), v∈Rnv \in \R^n, belong to a unique homothety class of convex sets, (2) K1K_1 and K2K_2 are generalized nn-simplices whose nn-dimensional intersections K1∩(v+K2)K_1 \cap (v + K_2), v∈Rnv \in \R^n, are homothetic to a unique generalized nn-simplex.

Author information

Contact details are reproduced from the original publication and may be historical.

Valeriu Soltan

Dept. of Mathematical Sciences, George Mason University, Fairfax, VA 22030, U.S.A.

vsoltan@gmu.edu

Keywords

  • Homothety
  • convex body
  • intersection
  • generalized simplex

Mathematics Subject Classification

52A20

References

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Suggested citation

V. Soltan. “A Characteristic Intersection Property of Generalized Simplices.” Journal of Convex Analysis 18 (2011), No. 2, 529–543. https://doi.org/10.68381/jca18034

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Published by Heldermann Verlag, 2011. Rights now held by Banach Press LLC.