Cremona Convexity, Frame Convexity, and a Theorem of Santal Institutional Repository Document

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.

authors

• Sottile, Frank

author list (cited authors)

• Goodman, J. E., Holmsen, A., Pollack, R., Ranestad, K., & Sottile, F.

citation count

• 0

complete list of authors

• Goodman, Jacob E||Holmsen, Andreas||Pollack, Ricky||Ranestad, Kristian||Sottile, Frank

Book Title

• arXiv

publication date

• September 2004

keywords

• 52a35; 14m15, 14m12
• Math.ag
• Math.mg

Digital Object Identifier (DOI)

• 10.48550/arxiv.math/0409219