2022/09/28 by Stefan Steinerberger, Steinerberger, Stefan
Computer Science · Mathematics · #Advanced Mathematical Modeling in Engineering #Classical Analysis and ODEs (math.CA) #Differential Geometry (math.DG) #FOS: Mathematics #Functional Analysis (math.FA) #Holomorphic and Operator Theory #Metric Geometry (math.MG) #Nonlinear Partial Differential Equations
paper · pdf · doi:10.48550/arxiv.2209.14153
openalex publication_date 2022/09/28 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
A property of smooth convex domains Ω⊂ ℝn is that if two points on the boundary x, y ∈ ∂ Ω are close to each other, then their normal vectors n(x), n(y) point roughly in the same direction and this direction is almost orthogonal to x-y (for `nearby' x and y). We prove there exists a constant cn > 0 such that if Ω⊂ ℝn is a bounded domain with C1-boundary ∂ Ω, then ∫∂ Ω× ∂ Ω \frac|⟨ n(x), y - x ⟩ ⟨ y - x, n(y) ⟩ | ‖x - y‖n+1~d σ(x) dσ(y) ≥ cn |∂ Ω| and equality occurs if and only if the domain Ω is convex.