Convex functions on real Banach spaces

Part of the Lecture Notes in Mathematics book series (LNM, volume 1364)


The letter E will always denote a real Banach space, D will be a nonempty open convex subset of E and ƒ will be a convex function on D. That is, ƒ: DR satisfies
$$ f\left[ {tx + \left( {1 - t} \right)y} \right] \leqslant tf\left( x \right) + \left( {1 - t} \right)f\left( y \right) $$
whenever x, yD and 0 < t < 1. If equality always holds, ƒ is said to be affine. A function ƒ: DR is said to be concave if — ƒ is convex. We will be studying the differentiability properties of such functions, assuming, in the beginning, that they are continuous. (The important case of lower semicontinuous convex functions is considered in Sec. 3.)


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Copyright information

© Springer-Verlag Berlin Heidelberg 1993

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