Sublattices of lattices of convex subsets of vector spaces

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For a left vector space V over a totally ordered division ring F, let Co(V) denote the lattice of convex subsets of V. We prove that every lattice L can be embedded into Co(V) for some left F-vector space V. Furthermore, if L is finite lower bounded, then V can be taken finite-dimensional, and L embeds into a finite lower bounded lattice of the form Co(V,Z)={X∩ Z | X∈ Co(V)}, for some finite subset Z of V. In particular, we obtain a new universal class for finite lower bounded lattices.