EQUILATERAL SETS IN UNIFORMLY SMOOTH BANACH SPACES. Issue 1 (January 2014)

• Record Type: Journal Article
• Authors: Freeman, D.; Odell, E.; Sari, B.; Schlumprecht, Th.
• Is Part Of: Mathematika. Volume 60:Issue 1(2014)
• Page Start: 219
• Page End: 231
• Publication Date: 2014-01
• Subjects: Mathematics -- Periodicals; 510.5
• DOI: 10.1112/S0025579313000260
• Languages: English
• ISSNs: 0025-5793

Abstract

<abstract abstract-type="normal"> <title>Abstract</title> <p>Let <inline-formula><alternatives><inline-graphic xlink:href="ark:/27927/pgg43k63x7h" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><tex-math><![CDATA[$X$]]></tex-math></alternatives></inline-formula> be an infinite-dimensional uniformly smooth Banach space. We prove that <inline-formula><alternatives><inline-graphic xlink:href="ark:/27927/pgg43k63z1q" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><tex-math><![CDATA[$X$]]></tex-math></alternatives></inline-formula> contains an infinite equilateral set. That is, there exist a constant <inline-formula><alternatives><inline-graphic xlink:href="ark:/27927/pgg43k640zm" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><tex-math><![CDATA[$\lambda \gt 0$]]></tex-math></alternatives></inline-formula> and an infinite sequence <inline-formula><alternatives><inline-graphic xlink:href="ark:/27927/pgg43k63wf9" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><tex-math><![CDATA[$\mathop{({x}_{i} )}\nolimits_{i= 1}^{\infty } \subset X$]]></tex-math></alternatives></inline-formula> such that <inline-formula><alternatives><inline-graphic xlink:href="ark:/27927/pgg43k63w1p" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><tex-math><![CDATA[$\Vert {x}_{i} - {x}_{j} \Vert = \lambda $]]></tex-math></alternatives></inline-formula> for all <inline-formula><alternatives><inline-graphic xlink:href="ark:/27927/pgg43k63zkj" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><tex-math><![CDATA[$i\not = j$]]></tex-math></alternatives></inline-formula>.</p> </abstract>