Cremona convexity, frame convexity and a theorem of Santaló
Jacob Eli Goodman, Andreas F. Holmsen, Richard Pollack, Kristian Ranestad, Frank Sottile · Advances in Geometry · 2006
Abstract In 1940, Luis Santaló proved a Helly-type theorem for line transversals to boxes in ℝ d . An analysis of his proof reveals a convexity structure for ascending lines in ℝ d that is isomorphic to the ordinary notion of convexity in a convex subset of ℝ2d−2. This isomorphism is through a Cremona transformation on the Grassmannian of lines in ℙ d , which enables a precise description of the convex hull and affine span of up to d ascending lines: the lines in such an affine span turn out to be the rulings of certain classical determinantal varieties. Finally, we relate Cremona convexity to a new convexity structure that we call frame convexity, which extends to arbitrary-dimensional flats in ℝ d .