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/pghjkq0qpf" 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/pghjkq0rnd" 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/pghjkq0tsf" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /> <![CDATA[λ 0]]> </alternatives> and an infinite sequence <alternatives><inline-graphic xlink:href="ark:/27927/pghjkq0pmx" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /> <![CDATA[(x_i ) olimits_i= 1^∞ X]]> </alternatives> such that <alternatives><inline-graphic xlink:href="ark:/27927/pghjkq0p35" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /> <![CDATA[x_i - x_j = λ ]]> </alternatives> for all <alternatives><inline-graphic xlink:href="ark:/27927/pghjkq0s65" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /> <![CDATA[i= j]]> </alternatives>.