Geodesic
Sublattices of Lattices of Convex Subsets of Vector Spaces
Algebra i logika, Tome 43 (2004) no. 3, pp. 261-290.

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Let ${\mathbf{Co}}(V)$ be a lattice of convex subsets of a vector space $V$ over a totally ordered division ring ${\mathbb{F}}$. We state that every lattice $L$ can be embedded into ${\mathbf{Co}}(V)$, for some space $V$ over ${\mathbb{F}}$. Furthermore, if $L$ is finite lower bounded, then $V$ can be taken finite-dimensional; in this case $L$ also embeds into a finite lower bounded lattice of the form ${\mathbf{Co}}(V,\Omega)=\{X\cap\Omega \mid X\in {\mathbf{Co}}(V)\}$, for some finite subset $\Omega$ of $V$. This result yields, in particular, a new universal class of finite lower bounded lattices.
Keywords: lattice of convex subsets of a vector space, finite lower bounded lattice. 
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F. Wehrung; M. V. Semenova. Sublattices of Lattices of Convex Subsets of Vector Spaces. Algebra i logika, Tome 43 (2004) no. 3, pp. 261-290. http://geodesic.mathdoc.fr/item/AL_2004_43_3_a0/

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