%0 Journal Article
%T Cremona Convexity, Frame Convexity, and a Theorem of Santal¨®
%A Jacob E. Goodman
%A Andreas Holmsen
%A Ricky Pollack
%A Kristian Ranestad
%A Frank Sottile
%J Mathematics
%D 2004
%I arXiv
%X In 1940, Luis Santal\'o 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 R^{2d-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.
%U http://arxiv.org/abs/math/0409219v2