J. E Goodman, A Holmsen, R Pollack, K Ranestad, F Sottile
Cremona convexity, frame convexity and a theorem of Santaló
In 1940, Luis Santaló proved a Helly-type theorem for line transversals to boxes in ℝd. An analysis of his proof reveals a convexity structure for ascending lines in ℝd 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 ℙd, 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 ℝd.
Advances in Geometry, Walter de Gruyter
Print ISSN: 1615-715X
Volume: 6, 03/2006
Pages: 301 - 321
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