The convexity behind Santalo's Helley-type Theorem </HEAD> <BODY BGCOLOR="#FFFFFF"> <HR><img hspace=10 align=center src="../images/ams-logo.gif"> <BR clear=all><HR> <TABLE> <TR> <TD> <A HREF="http://www.ams.org/amsmtgs/2120_program.html">AMS Eastern Sectional Meeting in <b>Annandale-on-Hudson, NY</b></A><BR> 8-9 October 2005 <P> <B>Title:</B>The convexity behind Santaló's Helley-type Theorem.<BR> <B>Session Name:</B> Special Session on Geometric Transversal Theory <P> <B>Author:</B> A. Holmsen<BR> <B>Author:</B> E. Goodman<BR> <B>Author:</B> R. Pollack<BR> <B>Author:</B> K. Ranestad<BR> <B>Author:</B> F. Sottile<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. </TD><TD><img src="../../abstracts/images/Santalo.gif"></TD></TR></TABLE> <BR><BR><HR><BR> </BODY> </HTML>