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In 1940, Luis Santaló 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 2 d −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  .

Cremona convexity, frame convexity and a theorem of Santaló