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RESEARCH PRODUCT
Improved Bounds for Hermite–Hadamard Inequalities in Higher Dimensions
Barbara BrandoliniRobert G. SmitsThomas BeckSimon LarsonJeffrey J. LangfordKrzysztof BurdzyStefan SteinerbergerAntoine Henrotsubject
Pure mathematicsInequalitymedia_common.quotation_subject01 natural sciencesConvexitysymbols.namesakeMathematics - Metric GeometrySettore MAT/05 - Analisi MatematicaHadamard transformHermite–Hadamard inequality0103 physical sciencesClassical Analysis and ODEs (math.CA)FOS: Mathematics[MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP]Hermite-Hadamard inequality subharmonic functions convexity.0101 mathematicsComputingMilieux_MISCELLANEOUSsubharmonic functionsmedia_commonMathematicsSubharmonic functionHermite polynomialsconvexity010102 general mathematicsMetric Geometry (math.MG)Functional Analysis (math.FA)Mathematics - Functional AnalysisMSC : 26B25 28A75 31A05 31B05 35B50Mathematics::LogicHermite-Hadamard inequalityDifferential geometryMathematics - Classical Analysis and ODEsFourier analysissymbols010307 mathematical physicsGeometry and Topologydescription
Let $\Omega \subset \mathbb{R}^n$ be a convex domain and let $f:\Omega \rightarrow \mathbb{R}$ be a positive, subharmonic function (i.e. $\Delta f \geq 0$). Then $$ \frac{1}{|\Omega|} \int_{\Omega}{f dx} \leq \frac{c_n}{ |\partial \Omega| } \int_{\partial \Omega}{ f d\sigma},$$ where $c_n \leq 2n^{3/2}$. This inequality was previously only known for convex functions with a much larger constant. We also show that the optimal constant satisfies $c_n \geq n-1$. As a byproduct, we establish a sharp geometric inequality for two convex domains where one contains the other $ \Omega_2 \subset \Omega_1 \subset \mathbb{R}^n$: $$ \frac{|\partial \Omega_1|}{|\Omega_1|} \frac{| \Omega_2|}{|\partial \Omega_2|} \leq n.$$
| year | journal | country | edition | language |
|---|---|---|---|---|
| 2019-07-13 | The Journal of Geometric Analysis |