Constructive approximation of a ball by polytopes
G. Beer defined the visibility function of a set S and proved its continuity in the interior of S. It is proved here that the visibility function of a planar Jordan domain is continuous precisely at the cone points of the boundary of S.
In his work on log-concavity of multiplicities, Okounkov showed in passing that one could associate a convex body to a linear series on a projective variety, and then use convex geometry to study such linear systems. Although Okounkov was essentially working in the classical setting of ample line bundles, it turns out that the construction goes through for an arbitrary big divisor. Moreover, this viewpoint renders transparent many basic facts about asymptotic invariants of linear series, and opens...