A note on subgaussian estimates for linear functionals on convex bodies

We give an alternative proof of a recent result of Klartag on the existence of almost subgaussian linear functionals on convex bodies. If K is a convex body in {\mathbb R}^n with volume one and center of mass at the origin, there exists x\neq 0 such that |{ y\in K:|\langle y,x\rangle |\geq t\|\langl...

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Bibliographic Details
Published in:Proceedings of the American Mathematical Society 2007-08, Vol.135 (8), p.2599-2606
Main Authors: Giannopoulos, A., Pajor, A., Paouris, G.
Format: Article
Language:English
Subjects:
Asymptotic theory
Center of mass
Convex and discrete geometry
Exact sciences and technology
General mathematics
General, history and biography
Geometry
Mathematical constants
Mathematical inequalities
Mathematical Physics
Mathematical theorems
Mathematics
Sciences and techniques of general use
Unit ball
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Summary:We give an alternative proof of a recent result of Klartag on the existence of almost subgaussian linear functionals on convex bodies. If K is a convex body in {\mathbb R}^n with volume one and center of mass at the origin, there exists x\neq 0 such that |{ y\in K:|\langle y,x\rangle |\geq t\|\langle·,x\rangle\|_1\}|\leq\exp (-ct^2/\log^2(t+1)) for all t\geq 1, where c>0 is an absolute constant. The proof is based on the study of the L_q--centroid bodies of K. Analogous results hold true for general log-concave measures.
ISSN:0002-9939
1088-6826
DOI:10.1090/S0002-9939-07-08778-3