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On lattices of convex sets in R^n

  • Author(s): Bergman, George M
  • et al.
Abstract

Properties of several sorts of lattices of convex subsets of Rn are examined. The lattice of convex sets containing the origin turns out, for n > 1, to satisfy a set of identities strictly between those of the lattice of all convex subsets of Rn and the lattice of all convex subsets of Rn-1. The lattices of arbitrary, of open bounded, and of compact convex sets in Rn all satisfy the same identities, but the last of these is join-semidistributive, while for n > 1 the first two are not. The lattice of relatively convex subsets of a fixed set S subset of Rn satisfies some, but in general not all of the identities of the lattice of "genuine" convex subsets of Rn.

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