Cremona Convexity, Frame Convexity, and a Theorem of Santal\'o
Abstract
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.
Keywords
Cite
@article{arxiv.math/0409219,
title = {Cremona Convexity, Frame Convexity, and a Theorem of Santal\'o},
author = {Jacob E. Goodman and Andreas Holmsen and Ricky Pollack and Kristian Ranestad and Frank Sottile},
journal= {arXiv preprint arXiv:math/0409219},
year = {2010}
}
Comments
18 pages, 4 .eps figures Corrected statement of Helly's Theorem in the Introduction