  A special type of Gagliardo-Nirenberg-Sobolev (GNS) inequalities in
$\mathbb{R}^d$ has played a key role in several proofs of Lieb-Thirring
inequalities. Recently, a need for GNS inequalities in convex domains of
$\mathbb{R}^d$, in particular for cubes, has arised. The purpose of this
manuscript is two-fold. First we prove a GNS inequality for convex domains,
with explicit constants which depend on the geometry of the domain. Later,
using the discrete version of Rumin's method, we prove GNS inequalities on
cubes with improved constants.
