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The Convex Hull of Extremal Vector‐Valued Continuous Functions
Author(s) -
MenaJurado J. F.,
NavarroPascual J. C.
Publication year - 1995
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/27.5.473
Subject(s) - mathematics , unit sphere , convex hull , dimension (graph theory) , combinatorics , regular polygon , ball (mathematics) , space (punctuation) , convex function , continuous function (set theory) , function (biology) , pure mathematics , mathematical analysis , geometry , linguistics , philosophy , evolutionary biology , biology
Let T be a completely regular space and X a strictly convex n ‐dimensional real space. We prove that every continuous function from T into the closed unit ball of X can be expressed as an average of eight continuous functions from T into the sphere of X if and only if dim ( T ) ⩽ n −1, where dim( T ) denotes the covering dimension of T . The proof we give can be used to prove the same fact, without hypotheses on T , when X is infinite‐dimensional, although in this case it has been proved recently that a better result can be obtained.