Local positivity of line bundles on smooth toric varieties and Cayley polytopes

For a smooth projective variety X and a line bundle L on X there are various notions for measuring the local positivity of L at a point x ∈ X . Two such measures are the dimension of the k-osculating space at x and the Seshadri constat at x. The precise nature of the interplay between these two notions is in general an open question. Our main result gives a partial answer to this question for smooth toric varieties, by showing that these notions characterize Cayley sums.