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Erschienen in:
2022 | OriginalPaper | Buchkapitel
A Characterization for the Validity of the Hermite–Hadamard Inequality on a Simplex
verfasst von : Allal Guessab
Erschienen in: Approximation and Computation in Science and Engineering
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Abstract
We consider the d-dimensional Hermite–Hadamard inequality
$$\displaystyle {} \frac {1}{\left |S\right |} \int _{S}f({\boldsymbol x}) \, d{\boldsymbol x} \leq Q^{\text{ tra}}(f):= \frac {1}{\left |\partial S\right |} \int \limits _{\partial S}f({\boldsymbol x})d\gamma . $$
(1)
Here f is a convex function defined on a simplex \(S\subset \mathbb {R}^d, (d\in \mathbb {N}).\) We give necessary and sufficient conditions on S for the validity of (1). More specifically, we establish that (1) holds if and only if S is an equiareal simplex. We will give two proofs of this result:
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The first proof is based on Green’s identity. Here, in addition to the convexity requirement, the C1-regularity assumption is necessary.
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In the second proof, the convexity is only required.
A series of equivalent criteria for validity of (1) is simply reformulated in terms of coincidences of certain simplex centers.