(Normal)
A lattice polytope \(P \subseteq M_\RR\) is normal if \[ (kP \intersect M) + (\ell P \intersect M) = (k+\ell)P \intersect M \quad \text{for all } k, \ell \geq 1 , \] equivalently \(k \cdot (P \intersect M) = (kP) \intersect M\) for all \(k \geq 1\), equivalently \((P \intersect M) \times \ts{1}\) generates the semigroup \(C(P) \intersect (M \times \ZZ)\), where \(C(P) \da \Cone(P \times \ts{1})\).
(Normality and dilation)
Normal implies very ample.
If \(P\) is full-dimensional with \(\dim P \geq 2\), then \(kP\) is normal for every \(k \geq \dim P - 1\).
(Consequences for very ampleness)
For a full-dimensional lattice polytope \(P\) of dimension \(d\ge2\), the proposition gives that \(kP\) is normal, hence very ample, for every \(k\ge d-1\). For \(d=2\) this holds for \(k=1\): every lattice polygon is normal, so every ample divisor on a complete toric surface is very ample. For \(d=3\), \(kP\) is normal for every \(k\ge2\), so a lattice polytope of dimension three that is not very ample can occur only at \(k=1\).
The toric variety \(X_P\) is \(\Proj\) of the semigroup ring \(\CC[C(P)\intersect(M\times\ZZ)]\), graded by the last coordinate. So \(P\) is normal if and only if this ring is generated in degree \(1\), that is, if and only if the embedding of \(X_P\) by the characters in \(P\intersect M\) is projectively normal.