When Is a Set of Lines in Space Convex?

The Definition of a Convex Set In Rd, a set S of points is convex if the line segment joining any two points of S lies completely within S (Figure 1). The purpose of this article is to describe a recent extension of this concept of convexity to the Grassmannian and to discuss its connection with some other ideas in geometry. More specifically, the extension is to the so-called “affine Grassmannian” G′ k,d, the open manifold that parametrizes all the k-dimensional flats (translates of linear subspaces) in Rd. In other words, rather than convex sets of points, we will be talking about convex sets of lines, for example, or of planes. Much of the material that this article deals with is based on joint work of the author’s with Richard Pollack [7, 8], as well as with Rephael Wenger and others [3, 10]. Which properties of convex point sets would we expect to hold also for convex sets of k-flats? The basic setup for point sets is this. To any set S of points is associated a set conv S containing S , called its convex hull (Figure 2), that satisfies: 1. monotonicity: S ⊂ T =⇒ conv S ⊂ conv T. 2. idempotence: conv (conv S) = conv S.