  We study asymptotic invariants of linear series on surfaces with the help of
Newton-Okounkov polygons. Our primary aim is to understand local positivity of
line bundles in terms of convex geometry. We work out characterizations of
ample and nef line bundles in terms of their Newton-Okounkov bodies, treating
the infinitesimal case as well. One of the main results is a description of
moving Seshadri constants via infinitesimal Newton-Okounkov polygons. As an
illustration of our ideas we reprove results of Ein-Lazarsfeld on Seshadri
constants on surfaces.
