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Trees, Gâteaux norms and a problem of Haydon
Author(s) -
Smith Richard J.
Publication year - 2007
Publication title -
bulletin of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 2.396
H-Index - 48
eISSN - 1469-2120
pISSN - 0024-6093
DOI - 10.1112/blms/bdl022
Subject(s) - mathematics , norm (philosophy) , regular polygon , tree (set theory) , convex function , pure mathematics , combinatorics , law , geometry , political science
We give a sufficient condition for the existence of an equivalent Gâteaux smooth norm on 0 (ϒ), where ϒ is a tree. Using this, we prove that if 0 (ϒ) admits an equivalent strictly convex norm, then it admits an equivalent Gâteaux smooth norm. This resolves an open problem from Haydon's study of trees in renorming theory.