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Geometry of the $ L_q$-centroid bodies of an isotropic log-concave measure


Authors: Apostolos Giannopoulos, Pantelis Stavrakakis, Antonis Tsolomitis and Beatrice-Helen Vritsiou
Journal: Trans. Amer. Math. Soc. 367 (2015), 4569-4593
MSC (2010): Primary 52A23; Secondary 46B06, 52A40, 60D05
DOI: https://doi.org/10.1090/S0002-9947-2015-06177-7
Published electronically: February 3, 2015
MathSciNet review: 3335394
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Abstract: We study some geometric properties of the $ L_q$-centroid bodies $ Z_q(\mu )$ of an isotropic log-concave measure $ \mu $ on $ {\mathbb{R}}^n$. For any $ 2\leqslant q\leqslant \sqrt {n}$ and for $ \varepsilon \in (\varepsilon _0(q,n),1)$ we determine the inradius of a random $ (1-\varepsilon )n$-dimensional projection of $ Z_q(\mu )$ up to a constant depending polynomially on $ \varepsilon $. Using this fact we obtain estimates for the covering numbers $ N(\sqrt {\smash [b]{q}}B_2^n,tZ_q(\mu ))$, $ t\geqslant 1$, thus showing that $ Z_q(\mu )$ is a $ \beta $-regular convex body. As a consequence, we also get an upper bound for $ M(Z_q(\mu ))$.


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Additional Information

Apostolos Giannopoulos
Affiliation: Department of Mathematics, University of Athens, Panepistimioupolis 15784, Athens, Greece
Email: apgiannop@math.uoa.gr

Pantelis Stavrakakis
Affiliation: Department of Mathematics, University of Athens, Panepistimioupolis 15784, Athens, Greece
Email: pantstav@yahoo.gr

Antonis Tsolomitis
Affiliation: Department of Mathematics, University of the Aegean, Karlovassi 83200, Samos, Greece
Email: antonis.tsolomitis@gmail.com

Beatrice-Helen Vritsiou
Affiliation: Department of Mathematics, University of Athens, Panepistimioupolis 15784, Athens, Greece
Address at time of publication: Department of Mathematics, University of Michigan, Ann Arbor, Michigan 48109
Email: bevritsi@math.uoa.gr, vritsiou@umich.edu

DOI: https://doi.org/10.1090/S0002-9947-2015-06177-7
Received by editor(s): January 21, 2013
Published electronically: February 3, 2015
Article copyright: © Copyright 2015 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.