Convex functions on real Banach spaces

Abstract

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: D → R 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, y ∈ D and 0 < t < 1. If equality always holds, ƒ is said to be affine. A function ƒ: D → R 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.)