  We construct two families of Dantzig figures, which are $(d,2d)$-polytopes
with an antipodal vertex pair, from convex hulls of initial subsets for the
graded lexicographic (grlex) and graded reverse lexicographic (grevlex) orders
on $\mathbb{Z}^{d}_{\geq 0}$. These two polytopes have the same number of
vertices, $\mathcal{O}(d^{2})$, and the same number of edges,
$\mathcal{O}(d^{3})$, but are not combinatorially equivalent. We provide an
explicit description of the vertices and the facets for both families and
describe their graphs along with analyzing their basic properties such as the
radius, diameter, existence of Hamiltonian circuits, and chromatic number.
Moreover, we also analyze the edge expansions of these graphs.
