On lattices of convex sets in R

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 R. 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 ⊆ Rn satisfies some, but in general not all of the identities of the lattice of “genuine” convex subsets of Rn. 1. Notation, conventions, remarks For S a subset of R, the convex hull of S will be denoted c.h.(S) = {Σmi=1 λi pi | m ≥ 1, pi ∈ S, λi ∈ [0, 1], Σλi = 1}. (1) When S is written as a list of elements “{ ··· },” we generally simplify “c.h.({ ··· })” to “c.h.( ··· )”. Conv(R) will denote the lattice of all convex subsets of R; its lattice operations are x ∧ y = x ∩ y, x ∨ y = c.h.(x ∪ y). (2) In any lattice, if a finite family of elements yi has been specified, then an expression such as ∧ i yi will denote the meet over the full range of the index i, and similarly for joins. Likewise, if we write something like ∧ j 6=i yj where i has been quantified outside this expression, then the meet will be over all values of j in the indexing family other than i. If L is any lattice and x an element of L, or, more generally, of an overlattice of L, we define the sublattice L≥x = {y ∈ L | y ≥ x}. (3) Presented by N. Zaguia. Received April 22, 2003; accepted in final form February 16, 2005. 2000 Mathematics Subject Classification: 06B20, 52A20; 06E10, 54H12.