A Characterization for the Validity of the Hermite–Hadamard Inequality on a Simplex

We consider the d-dimensional Hermite–Hadamard inequality 1 1 S ∫ S f ( x ) d x ≤ Q tra ( f ) : = 1 ∂ S ∫ ∂ S f ( x ) d γ . $$\displaystyle {} \frac {1}{\left |S\right |} \int _{S}f({\boldsymbol x}) \, d{\boldsymbol x} \leq Q^{\text{ tra}}(f):= \frac {1}{\left |\partial S\right |} \int \limits _{\partial S}f({\boldsymbol x})d\gamma . $$ Here f is a convex function defined on a simplex S ⊂ ℝ d , ( d ∈ ℕ ) . $$S\subset \mathbb {R}^d, (d\in \mathbb {N}).$$ 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: • The first proof is based on Green’s identity. Here, in addition to the convexity requirement, the C1-regularity assumption is necessary. • 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.