Sublattices of Lattices of Convex Subsets of Vector Spaces

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.