  Nicaise--Ottem introduced the notion of (stably) rational polytopes and
studied this using a combinatorial description of the motivic volume. In this
framework, we ask whether being non-stably rational is preserved under
inclusions. We prove this holds for a large class of polytopes, leading to a
combinatorial strategy for studying stable rationality of hypersurfaces in
toric varieties. As a result, we obtain new bounds for non-stably rational
hypersurface in projective space, improving the ones given by Schreieder when
the field has characteristic 0. We also obtain similar bounds for double covers
of projective space and some new classes of non-stably rational varieties in
products of projective space.
