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Optimality of CKP-inequality in the critical case


Author: Fuchang Gao
Journal: Proc. Amer. Math. Soc. 142 (2014), 909-914
MSC (2010): Primary 41A46, 47B06; Secondary 60G15
DOI: https://doi.org/10.1090/S0002-9939-2013-11825-3
Published electronically: November 26, 2013
MathSciNet review: 3148525
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Abstract: It is proved that the CKP inequality

$\displaystyle \sqrt {\log N({\rm cov}(T),2\varepsilon )} \preceq \frac 1\varepsilon \int _{\varepsilon /2}^\infty \sqrt {\log N(T,r)} dr$

is optimal in the critical case $ \log N(T,\varepsilon )=O(\varepsilon ^{-2}\vert\log \varepsilon \vert^{-2})$ as $ \varepsilon \to 0^+$.

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Additional Information

Fuchang Gao
Affiliation: Department of Mathematics, University of Idaho, Moscow, Idaho 83844-1103
Email: fuchang@uidaho.edu

DOI: https://doi.org/10.1090/S0002-9939-2013-11825-3
Keywords: Metric entropy, convex hull, CKP inequality, small ball probability
Received by editor(s): January 19, 2012
Received by editor(s) in revised form: April 5, 2012
Published electronically: November 26, 2013
Additional Notes: This work was partially supported by a grant from the Simons Foundation, No. 246211
Communicated by: Thomas Schlumprecht
Article copyright: © Copyright 2013 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.