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Integral inequality for concave function on convex set

Source: Schweitzer 2009

November 13, 2009
calculusintegrationinequalitiesfunctioninvariantreal analysisreal analysis unsolved

Problem Statement

Let P⊆Rm P\subseteq \mathbb{R}^m be a non-empty compact convex set and f: P\rightarrow \mathbb{R}_{ \plus{} } be a concave function. Prove, that for every ξ∈Rm \xi\in \mathbb{R}^m \int_{P}\langle \xi,x \rangle f(x)dx\leq \left[\frac {m \plus{} 1}{m \plus{} 2}\sup_{x\in P}{\langle\xi,x\rangle} \plus{} \frac {1}{m \plus{} 2}\inf_{x\in P}{\langle\xi,x\rangle}\right] \cdot\int_{P}f(x)dx.
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