2022/10/10 by Dylan Langharst, Langharst, Dylan, Putterman, Eli · 1 citation
Mathematics · Physics and Astronomy · #28A75 (Secondary) #52A39 #52A41 (Primary) #Advanced Differential Geometry Research #FOS: Mathematics #Functional Analysis (math.FA) #Mathematical Inequalities and Applications #Metric Geometry (math.MG) #Point processes and geometric inequalities
paper · pdf · doi:10.48550/arxiv.2210.04438
openalex publication_date 2022/10/10 · openalex created_date 2022/10/12 · openalex updated_date 2026/08/02
The inequality of Berwald is a reverse-Hölder like inequality for the pth average, p∈ (-1,∞), of a non-negative, concave function over a convex body in ℝn. We prove Berwald's inequality for averages of functions with respect to measures that have some concavity conditions, e.g. s-concave measures, s∈ ℝ. We also obtain equality conditions; in particular, this provides a new proof for the equality conditions of the classical inequality of Berwald. As applications, we generalize a number of classical bounds for the measure of the intersection of a convex body with a half-space and also the concept of radial means bodies and the projection body of a convex body.