Vol. 60, No. 2, 1975

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ISSN: 0030-8730
A representation theorem for real convex functions

Roy Martin Rakestraw

Vol. 60 (1975), No. 2, 165–168
Abstract

The Krein-Milman theorem is used to prove the following result. A nonnegative function f on [0,1] convex if, and only if, there exist nonnegative Borel measures μ1 and μ2 on [0,1] such that

       ∫ x     −1              ∫ 1
f (x) =  0 (1 − ξ)  (x− ξ)dμ1(ξ)+  x [1 − (x∕ξ)]dμ2(ξ),

for every x [0,1]. An example is given for which the representation is not unique.

Mathematical Subject Classification 2000
Primary: 26A51
Milestones
Received: 31 January 1974
Published: 1 October 1975
Authors
Roy Martin Rakestraw