  We introduce the property of convex normality of rational polytopes and give
a dimensionally uniform lower bound for the edge lattice lengths, guaranteeing
the property. As an application, we show that if every edge of a lattice
d-polytope P has lattice length at least 4d(d+1) then P is normal. This answers
in the positive a question raised in 2007. If P is a lattice simplex whose
edges have lattice lengths at least d(d+1) then P is even covered by lattice
parallelepipeds. For the approach developed here, it is necessary to involve
rational polytopes even for applications to lattice polytopes.
