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Convex Transitive Norms on Spaces of Continuous Functions
Author(s) -
Sánchez Félix Cabello
Publication year - 2005
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/s0024609304003649
Subject(s) - mathematics , unit sphere , norm (philosophy) , pure mathematics , convex hull , isometry (riemannian geometry) , convex function , combinatorics , regular polygon , geometry , political science , law
A norm on a Banach space X is called maximal when no equivalent norm has a larger group of isometries. If, besides this, there is no equivalent norm with the same isometries (apart from its scalar multiples), the norm is said to be uniquely maximal, which is equivalent to the convex‐transitivity of X : the convex hull of the orbits under the action of the isometry group on the unit sphere is dense in the unit ball of X . The main result of the paper is that the complex C 0 (Ω) is convex‐transitive in its natural supremum norm if Ω is a connected manifold (without boundary). As a complement, it is shown that if Ω is a connected manifold of dimension at least two, then the diameter norm is convex transitive on the corresponding space of real functions. 2000 Mathematics Subject Classification 46B15, 47B99