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In 2012, Gubeladze (Adv. Math. 2012) introduced the notion of 
$k$-convex-normal polytopes to show that integral polytopes all of whose edges are longer than 
$4d(d+1)$ have the integer decomposition property. In the first part of this paper we show that for lattice polytopes there is no diòerence between $k$- and 
$(k+1)$-convex-normality (for 
$k\ge 3$) and improve the bound to 
$2d(d+1)$. In the second part we extend the definition to pairs of polytopes. Given two rational polytopes 
$P$ and 
$\text{Q}$, where the normal fan of $P$ is a refinement of the normal fan of $\text{Q}$, if every edge 
${{e}_{P}}$ of $P$ is at least 
$d$ times as long as the corresponding face (edge or vertex) 
${{e}_{\text{Q}}}$ of $\text{Q}$, then 
$(P+\text{Q})\cap {{\mathbb{Z}}^{d}}=(P\cap {{\mathbb{Z}}^{d}})+(\text{Q}\cap {{\mathbb{Z}}^{d}})$.
