TY - JOUR

T1 - Intrinsic volumes of Sobolev balls with applications to Brownian convex hulls

AU - Kabluchko, Zakhar

AU - Zaporozhets, Dmitry

PY - 2016/1/1

Y1 - 2016/1/1

N2 - 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 S1, S2, C1, C2 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 (Formula Presented.) 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.

AB - 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 S1, S2, C1, C2 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 (Formula Presented.) 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.

KW - Brownian convex hulls

KW - Brownian zonoids

KW - Ellipsoids

KW - Gaussian processes

KW - Intrinsic volumes

KW - Lipschitz balls

KW - Mean width

KW - Sobolev balls

KW - Sudakov’s formula

KW - Tsirelson’s theorem

UR - http://www.scopus.com/inward/record.url?scp=84990914058&partnerID=8YFLogxK

U2 - 10.1090/tran/6628

DO - 10.1090/tran/6628

M3 - Article

AN - SCOPUS:84990914058

VL - 368

SP - 8873

EP - 8899

JO - Transactions of the American Mathematical Society

JF - Transactions of the American Mathematical Society

SN - 0002-9947

IS - 12

ER -