Home Competitions SCHWEITZER 2009 Problem 9
SCHWEITZER 2009 Problem 9
Let P ⊆ R m P\subseteq \mathbb{R}^m P ⊆ R m f : P → R + f: P\rightarrow \mathbb{R}_{ + } f : P → R + ξ ∈ R m \xi\in \mathbb{R}^m ξ ∈ R m ∫ P ⟨ ξ , x ⟩ f ( x ) d x ≤ [ m + 1 m + 2 sup x ∈ P ⟨ ξ , x ⟩ + 1 m + 2 inf x ∈ P ⟨ ξ , x ⟩ ] ⋅ ∫ P f ( x ) d x . \int_{P}\langle \xi,x \rangle f(x)dx\leq \left[\frac {m + 1}{m + 2}\sup_{x\in P}{\langle\xi,x\rangle} + \frac {1}{m + 2}\inf_{x\in P}{\langle\xi,x\rangle}\right] \cdot\int_{P}f(x)dx. ∫ P ⟨ ξ , x ⟩ f ( x ) d x ≤ [ m + 2 m + 1 x ∈ P sup ⟨ ξ , x ⟩ + m + 2 1 x ∈ P inf ⟨ ξ , x ⟩ ] ⋅ ∫ P f ( x ) d x .
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