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Cremona Convexity, Frame Convexity, and a Theorem of Santal\'o

Jacob E. Goodman·Andreas Holmsen·Ricky Pollack·Kristian Ranestad·Frank Sottile
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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.

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Jacob E. Goodman, Andreas Holmsen, Ricky Pollack, Kristian Ranestad, Frank Sottile. 2004-09-14. Cremona Convexity, Frame Convexity, and a Theorem of Santal\'o. https://arxiv.org/abs/math/0409219

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