"Convexity" of sets of lines

Abstract Goodman and Pollack (Goodman and Pollack, 1995; Goodman, 1998) have introduced a concept of convexity for sets of translates of linear subspaces in n -dimensional space; in particular, their definition applies to sets of lines in the plane. Recently, Rosenfeld (1995) proposed two ways of defining geometric properties (such as convexity) for sets of lines – one in terms of properties of the corresponding sets of points in Hough space, and the other in terms of point/line incidence. This paper compares the three definitions, as well as a fourth definition which is a weakened version of Rosenfeld's incidence definition. We show that the Hough definition is incomparable with the other three, and that the incidence definitions are incomparable with each other but imply the Goodman–Pollack definition; these last results also hold in 3-space.