Convexity for lines in Rd Frank Sottile

    Convex subsets in R^d play an important role in Discrete and Computational Geometry. A cornerstone of the theory of convexity is Helly's Theorem from 1923 which states that if there is a point in common to every d+1 (or fewer) members of a finite collection of convex sets in R^d, then the whole family has a point in common. The usefulness of such a basic numerical result led to a search for similar results for other objects besides points, for example line transversals to convex sets in R^d. While that is impossible in general, in 1940 Santalo proved a Helly-type result for line transversals to boxes in R^d.

    This talk will describe these foundational results in Convex Geometry and discuss how an analysis of Santalo's arguments leads to a new and very pleasing convexity structure on ascending lines in R^d, one that is related to some interesting geometry of the Grassmannian of lines. For example, the two pictures above are the convex hull of two lines in R³ (on the left) and the convex hull of three lines (on the right).