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Entropy of chord distribution of convex bodies


Author: Wenxue Xu
Journal: Proc. Amer. Math. Soc. 147 (2019), 3131-3141
MSC (2010): Primary 52A20, 52A22, 60D05
DOI: https://doi.org/10.1090/proc/14447
Published electronically: March 15, 2019
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Abstract: For a fixed convex body $ K$ in $ \mathbb{R}^n$, the chord distribution on the affine Grassmann manifold of lines is introduced and then the entropy of chord distribution is defined. Using the integral geometric method, we establish that the entropy of chord distribution of a convex body $ K$ attains its minimum if and only if $ K$ is a ball.


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Additional Information

Wenxue Xu
Affiliation: School of Mathematics and Statistics, Southwest University, Chongqing 400715, People’s Republic of China
Email: xwxjk@163.com

DOI: https://doi.org/10.1090/proc/14447
Keywords: Chord distribution, chord power integrals, chord entropy, integral geometric density, radial function.
Received by editor(s): May 22, 2018
Received by editor(s) in revised form: September 17, 2018
Published electronically: March 15, 2019
Additional Notes: The author was supported in part by the Fundamental Research Funds for the Central Universities (No. XDJK2017C056), Ph.D. Research Fundation of Southwest University (No. SWU115053), the Natural Science Foundation Project of CQ CSTC (No. cstc2016jcyjA0465) and NSFC (Nos. 11401486 and 11501185).
Communicated by: Deane Yang
Article copyright: © Copyright 2019 American Mathematical Society