Geometry of the 𝐿_{𝑞}-centroid bodies of an isotropic log-concave measure

Giannopoulos, Apostolos; Stavrakakis, Pantelis; Tsolomitis, Antonis; Vritsiou, Beatrice-Helen

doi:10.1090/S0002-9947-2015-06177-7

Geometry of the $L_q$-centroid bodies of an isotropic log-concave measure
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by Apostolos Giannopoulos, Pantelis Stavrakakis, Antonis Tsolomitis and Beatrice-Helen Vritsiou PDF

Trans. Amer. Math. Soc. 367 (2015), 4569-4593 Request permission

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 $N(\sqrt {[b]{q}}B_2^n,tZ_q(\mu ))$, $t\geqslant 1$, 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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