The convexity behind Santalo's Helley-type Theorem </HEAD> <BODY BGCOLOR="#FFFFFF"> <HR> <TABLE> <TR> <TD> <P> <B>Title:</B>The convexity behind Santaló's Helley-type Theorem.<BR> F. Sottile<BR> Mittagsseminar, 2.02.06.<BR><BR> <b>Abstract:</b><br> In 1940, Luis Santaló proved a Helly-type theorem for line transversals to boxes in <b>R</b><sup><I>d</I></sup>. An analysis of his proof reveals a convexity structure for ascending lines in <b>R</b><sup><I>d</I></sup> that is isomorphic to the ordinary notion of convexity in a convex subset of <b>R</b><sup>2<I>d</I>-2</sup>. This isomorphism is through a Cremona transformation on the Grassmannian of lines in <b>P</b><sup><I>d</I></sup>, which enables a precise description of the convex hull and affine span of up to <I>d</I> ascending lines: the lines in such an affine span turn out to be the rulings of certain classical determinantal varieties.<BR> This talk represents joint work with A. Holmsen, E. Goodman, R. Pollack, and K. Ranestad, and is based upon the preprint <A HREF="../../pdf/santalo.pdf"><FONT COLOR="FF00FF"><I>Cremona Convexity, Frame Convexity, and a Theorem of Santaló</I></FONT></A>, which will appear in <A HREF="http://www.degruyter.de/rs/278_3129_ENU_h.htm">Advances in Geometry</A>. </TD><TD><img src="../../abstracts/images/Santalo.gif"></TD></TR></TABLE> <BR><BR><HR><BR> </BODY> </HTML>