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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Applications of Grünbaum-type inequalities


Authors: Matthew Stephen and Vlad Yaskin
Journal: Trans. Amer. Math. Soc. 372 (2019), 6755-6769
MSC (2010): Primary 52A20, 52A39, 52A40
DOI: https://doi.org/10.1090/tran/7879
Published electronically: August 5, 2019
MathSciNet review: 4024537
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Abstract: Let $ 1\leq i \leq k < n$ be integers. We prove the following exact inequalities for any convex body $ K\subset \mathbb{R}^n$ with centroid at the origin and any $ k$-dimensional subspace $ E\subset \mathbb{R}^n$:

  $\displaystyle V_i \big ( K\cap E \big ) \geq \left ( \frac {i+1}{n+1} \right )^i \max _{x\in K} V_i \big ( ( K-x) \cap E \big ) ,$    
  $\displaystyle \widetilde {V}_i \big ( K\cap E \big ) \geq \left ( \frac {i+1}{n+1} \right )^i \max _{x\in K} \widetilde {V}_i \big ( ( K-x) \cap E \big ) ;$    

$ V_i$ is the $ i$th intrinsic volume, and $ \widetilde {V}_i$ is the $ i$th dual volume taken within $ E$. Our results are an extension of an inequality of M. Fradelizi, which corresponds to the case $ i=k$. Using the same techniques, we also establish extensions of ``Grünbaum's inequality for sections'' and ``Grünbaum's inequality for projections'' to dual volumes.

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Additional Information

Matthew Stephen
Affiliation: School of Mathematical Sciences, Tel Aviv University, Tel Aviv, 69978, Israel
Email: matthews@mail.tau.ac.il

Vlad Yaskin
Affiliation: Department of Mathematical & Statistical Sciences, University of Alberta, Edmonton, Alberta, T6G 2G1, Canada
Email: yaskin@ualberta.ca

DOI: https://doi.org/10.1090/tran/7879
Keywords: Convex body, centroid, sections, intrinsic volumes, dual volumes
Received by editor(s): September 15, 2018
Received by editor(s) in revised form: April 5, 2019, and April 28, 2019
Published electronically: August 5, 2019
Additional Notes: Both authors were supported in part by NSERC
Additionally, for the first author, this publication is part of a project that has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement No. 770127)
Article copyright: © Copyright 2019 American Mathematical Society