KTH/SU Mathematics Colloquium
05-11-09
Kristian Ranestad, University of Oslo
What is a convex set of lines?
Luis Santaló (1940) 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 given by a Cremona transformation on the Grassmannian
of lines in R^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.