Improved Bounds for Hermite-Hadamard Inequalities in Higher Dimensions

2019/07/13 by Beck, Thomas, Brandolini, Barbara, Burdzy, Krzysztof +5 · 1 citation
#Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Functional Analysis (math.FA) #Metric Geometry (math.MG)

paper · doi:10.48550/arxiv.1907.06122

Abstract

Let Ω⊂ ℝⁿ be a convex domain and let f:Ω→ ℝ be a positive, subharmonic function (i.e. Δf ≥ 0). Then (1)/(|Ω|) ∫_Ωf dx ≤ (c_n)/( |∂ Ω| ) ∫_{∂ Ω} f dσ, where c_n ≤ 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 ≥ n-1. As a byproduct, we establish a sharp geometric inequality for two convex domains where one contains the other Ω₂ ⊂ Ω₁ ⊂ ℝⁿ: (|∂ Ω₁|)/(|Ω₁|) (| Ω₂|)/(|∂ Ω₂|) ≤ n.

Cited by

• Geometric properties of optimizers for the maximum gradient of the torsion function

Related

• The Hermite-Hadamard inequality in higher dimensions
• A sharp multidimensional Hermite-Hadamard inequality
• A Dimension-Free Hermite-Hadamard Inequality via Gradient Estimates for the Torsion Function
• Some Hermite-Hadamard Type Inequalities For Harmonically (s,m)-convex functions in Second Sense
• Hermite-Hadamard type inequalities for harmonically convex functions on the co-ordinates