Geometric representation of classes of concave functions and duality

2021/12/27 by Grigory Ivanov, Ivanov, Grigory, Elisabeth M. Werner +1
Mathematics · #46T12 #Advanced Optimization Algorithms Research #FOS: Mathematics #Functional Analysis (math.FA) #Mathematical Inequalities and Applications #Point processes and geometric inequalities #Primary 52A23 #Secondary 52A40

paper · pdf · doi:10.48550/arxiv.2112.13881

openalex publication_date 2021/12/27 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Using a natural representation of a 1/s-concave function on ℝ^d as a convex set in ℝ^d+1, we derive a simple formula for the integral of its s-polar. This leads to convexity properties of the integral of the s-polar function with respect to the center of polarity. In particular, we prove that that the reciprocal of the integral of the polar function of a log-concave function is log-concave as a function of the center of polarity. Also, we define the Santaló regions for s-concave and log-concave functions and generalize the Santaló inequality for them in the case the origin is not the Santaló point.

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