EQUILATERAL SETS IN UNIFORMLY SMOOTH BANACH SPACES. Issue 1 (January 2014) - Freeman, D.; Odell, E.; Sari, B.; Schlumprecht, Th.
Abstract Let <alternatives><inline-graphic xlink:href="ark:/27927/pgg43k63x7h" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /> <![CDATA[$X$]]> </alternatives> be an infinite-dimensional uniformly smooth Banach space. We prove that <alternatives><inline-graphic xlink:href="ark:/27927/pgg43k63z1q" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /> <![CDATA[$X$]]> </alternatives> contains an infinite equilateral set. That is, there exist a constant <alternatives><inline-graphic xlink:href="ark:/27927/pgg43k640zm" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /> <![CDATA[$\lambda \gt 0$]]> </alternatives> and an infinite sequence <alternatives><inline-graphic xlink:href="ark:/27927/pgg43k63wf9" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /> <![CDATA[$\mathop{({x}_{i} )}\nolimits_{i= 1}^{\infty } \subset X$]]> </alternatives> such that <alternatives><inline-graphic xlink:href="ark:/27927/pgg43k63w1p" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /> <![CDATA[$\Vert {x}_{i} - {x}_{j} \Vert = \lambda $]]> </alternatives> for all <alternatives><inline-graphic xlink:href="ark:/27927/pgg43k63zkj" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /> <![CDATA[$i\not = j$]]> </alternatives>.