Document Type
Article
Publication Date
6-13-2024
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 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.
Keywords
logarithmically concave function, s-concave function, Santaló inequality
Language
English
Publication Title
Journal of Geometric Analysis
Grant
DMS-2103482
Rights
© The Author(s) 2024. This is an Open Access work distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0/) which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Creative Commons License
This work is licensed under a Creative Commons Attribution 4.0 International License.
Recommended Citation
Ivanov, G., Werner, E.M. Geometric Representation of Classes of Concave Functions and Duality. J Geom Anal 34, 260 (2024). https://doi.org/10.1007/s12220-024-01703-9
Manuscript Version
Final Publisher Version