  For any non-negative integer $k$ the $k$-th osculating dimension at a given
point $x$ of a variety $X$ embedded in projective space gives a measure of the
local positivity of order $k$ at that point. In this paper we show that a
smooth toric embedding having maximal $k$-th osculating dimension, but not
maximal $(k+1)$-th osculating dimension, at every point is associated to a
Cayley polytope of order $k$. This result generalises an earlier
characterisation by David Perkinson. In addition we prove that the above
assumptions are equivalent to requiring that the Seshadri constant is exactly
$k$ at every point of $X$, generalising a result of Atsushi Ito.
