On Gaussian Marginals of Uniformly Convex Bodies |
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Authors: | Emanuel Milman |
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Institution: | (1) Department of Mathematics, The Weizmann Institute of Science, Rehovot, 76100, Israel;(2) Present address: School of Mathematics, Institute for Advanced Study, Einstein Drive, Princeton, NJ 08540, USA |
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Abstract: | Recently, Bo’az Klartag showed that arbitrary convex bodies have Gaussian marginals in most directions. We show that Klartag’s
quantitative estimates may be improved for many uniformly convex bodies. These include uniformly convex bodies with power
type 2, and power type p>2 with some additional type condition. In particular, our results apply to all unit-balls of subspaces of quotients of L
p
for 1<p<∞. The same is true when L
p
is replaced by S
p
m
, the l
p
-Schatten class space. We also extend our results to arbitrary uniformly convex bodies with power type p, for 2≤p<4. These results are obtained by putting the bodies in (surprisingly) non-isotropic positions and by a new concentration
of volume observation for uniformly convex bodies.
Supported in part by BSF and ISF. |
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Keywords: | Central limit theorem Convex bodies Uniformly convex bodies |
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