This is an announcement for the paper "Intrinsic volumes of Sobolev balls" by Zakhar Kabluchko and Dmitry Zaporozhets.
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 volumes of 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 $$ V_k = \frac{\pi^{k/2}}{\Gamma\left(\frac 32 k +1 \right)}, \text{ respectively } V_k = \frac{\pi^{(k+1)/2}}{2\Gamma\left(\frac 32 k +\frac 32\right)}. $$ 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., to appear]. 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. Our proofs exploit Sudakov's and Tsirelson's theorems which establish a connection between the intrinsic volumes and the isonormal Gaussian process.
Archive classification: math.PR math.FA math.MG
Mathematics Subject Classification: Primary, 60D05, secondary, 60G15, 52A22
Remarks: 23 pages
Submitted from: sachar.k@gmx.net
The paper may be downloaded from the archive by web browser from URL
http://front.math.ucdavis.edu/1404.6113
or