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Intrinsic volumes of Sobolev balls with applications to Brownian convex hulls


Authors: Zakhar Kabluchko and Dmitry Zaporozhets
Journal: Trans. Amer. Math. Soc. 368 (2016), 8873-8899
MSC (2010): Primary 60D05; Secondary 60G15, 52A22
DOI: https://doi.org/10.1090/tran/6628
Published electronically: February 12, 2016
MathSciNet review: 3551592
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Abstract: A formula due to Sudakov relates the first intrinsic volume of a convex set in a Hilbert space to the maximum of the isonormal Gaussian process over this set. Using this formula we compute the first intrinsic volume of some infinite-dimensional convex compact sets including unit balls with respect to Sobolev-type seminorms and ellipsoids in the Hilbert space. We relate the distribution of the random one-dimensional projections of these sets to the distributions $ S_1,S_2,C_1,C_2$ studied by Biane, Pitman, Yor [Bull. AMS 38 (2001)]. We show that the $ k$-th intrinsic volume of the set of all functions on $ [0,1]$ which have Lipschitz constant bounded by $ 1$ and which vanish at 0 (respectively, which have vanishing integral) is given by

$\displaystyle V_k = \frac {\pi ^{k/2}}{\Gamma \left (\frac 32 k +1 \right )},$$\displaystyle \text { respectively } V_k = \frac {\pi ^{(k+1)/2}}{2\Gamma \left (\frac 32 k +\frac 32\right )}. $

This is related to the results of Gao and Vitale [Discrete Comput. Geom. 26 (2001); Elect. Comm. Probab. 8 (2003)], who considered a similar question for functions with a restriction on the total variation instead of the Lipschitz constant. Using the results of Gao and Vitale we give a new proof of the formula for the expected volume of the convex hull of the $ d$-dimensional Brownian motion which is due to Eldan [Elect. J. Probab. 19 (2014)]. Additionally, we prove an analogue of Eldan's result for the Brownian bridge. Similarly, we show that the results on the intrinsic volumes of the Lipschitz balls can be translated into formulae for the expected volumes of zonoids (Aumann integrals) generated by the Brownian motion and the Brownian bridge. Also, these results have discrete versions for Gaussian random walks and bridges. Our proofs exploit Sudakov's and Tsirelson's theorems which establish a connection between the intrinsic volumes and the isonormal Gaussian process.

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Additional Information

Zakhar Kabluchko
Affiliation: Institute of Stochastics, Ulm University, Helmholtzstrasse 18, 89069 Ulm, Germany
Address at time of publication: Institute of Statistics, Orléans-Ring 10, 48149 Münster, Germany
Email: zakhar.kabluchko@uni-ulm.de, zakhar.kabluchko@uni-muenster.de

Dmitry Zaporozhets
Affiliation: St. Petersburg Department of Steklov Institute of Mathematics, Fontanka 27, 191011 St. Petersburg, Russia
Email: zap1979@gmail.com

DOI: https://doi.org/10.1090/tran/6628
Keywords: Intrinsic volumes, Gaussian processes, mean width, Sobolev balls, ellipsoids, Lipschitz balls, Brownian convex hulls, Brownian zonoids, Sudakov's formula, Tsirelson's theorem
Received by editor(s): May 11, 2014
Received by editor(s) in revised form: November 25, 2014
Published electronically: February 12, 2016
Additional Notes: The second author was supported by RFBR, grant 13-01-00256, and by CRC 701 “Spectral Structures and Topological Methods in Mathematics”, Bielefeld
Article copyright: © Copyright 2016 American Mathematical Society

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