Cremona convexity, frame convexity and a theorem of Santalo

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In 1940, Luis Santalo 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 R2d-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 in R-d.
Publisher
WALTER DE GRUYTER CO
Issue Date
2006
Language
English
Article Type
Article
Citation

ADVANCES IN GEOMETRY, v.6, no.2, pp.301 - 321

ISSN
1615-715X
DOI
10.1515/ADVGEOM.2006.018
URI
http://hdl.handle.net/10203/88732
Appears in Collection
MA-Journal Papers(저널논문)
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