## Intrinsic volumes of Sobolev balls with applications to Brownian convex hulls

HTML articles powered by AMS MathViewer

- by Zakhar Kabluchko and Dmitry Zaporozhets PDF
- Trans. Amer. Math. Soc.
**368**(2016), 8873-8899 Request permission

## 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 \[ V_k = \frac {\pi ^{k/2}}{\Gamma \left (\frac 32 k +1 \right )}, \text { respectively } V_k = \frac {\pi ^{(k+1)/2}}{2\Gamma \left (\frac 32 k +\frac 32\right )}. \] 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.*

**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.

## References

- Glen Baxter,
*A combinatorial lemma for complex numbers*, Ann. Math. Statist.**32**(1961), 901–904. MR**126290**, DOI 10.1214/aoms/1177704985 - Philippe Biane and Gérard Letac,
*The mean perimeter of some random plane convex sets generated by a Brownian motion*, J. Theoret. Probab.**24**(2011), no. 2, 330–341. MR**2795042**, DOI 10.1007/s10959-009-0272-0 - Philippe Biane, Jim Pitman, and Marc Yor,
*Probability laws related to the Jacobi theta and Riemann zeta functions, and Brownian excursions*, Bull. Amer. Math. Soc. (N.S.)**38**(2001), no. 4, 435–465. MR**1848256**, DOI 10.1090/S0273-0979-01-00912-0 - Vladimir I. Bogachev,
*Gaussian measures*, Mathematical Surveys and Monographs, vol. 62, American Mathematical Society, Providence, RI, 1998. MR**1642391**, DOI 10.1090/surv/062 - Yu. D. Burago and V. A. Zalgaller,
*Geometric inequalities*, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 285, Springer-Verlag, Berlin, 1988. Translated from the Russian by A. B. Sosinskiĭ; Springer Series in Soviet Mathematics. MR**936419**, DOI 10.1007/978-3-662-07441-1 - Simone Chevet,
*Processus Gaussiens et volumes mixtes*, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete**36**(1976), no. 1, 47–65 (French). MR**426120**, DOI 10.1007/BF00533208 - Ronen Eldan,
*Volumetric properties of the convex hull of an $n$-dimensional Brownian motion*, Electron. J. Probab.**19**(2014), no. 45, 34. MR**3210546**, DOI 10.1214/EJP.v19-2571 - Fuchang Gao,
*The mean of a maximum likelihood estimator associated with the Brownian bridge*, Electron. Comm. Probab.**8**(2003), 1–5. MR**1961284**, DOI 10.1214/ECP.v8-1064 - F. Gao and R. A. Vitale,
*Intrinsic volumes of the Brownian motion body*, Discrete Comput. Geom.**26**(2001), no. 1, 41–50. MR**1832728**, DOI 10.1007/s00454-001-0023-1 - D. N. Zaporozhets and Z. Kabluchko,
*Random determinants, mixed volumes of ellipsoids, and zeros of Gaussian random fields*, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI)**408**(2012), no. Veroyatnost′i Statistika. 18, 187–196, 327 (Russian, with English and Russian summaries); English transl., J. Math. Sci. (N.Y.)**199**(2014), no. 2, 168–173. MR**3032216**, DOI 10.1007/s10958-014-1844-9 - J. Kampf,
*Das Parallelvolumen und abgeleitete Funktionale*, PhD Thesis, Karlsruhe Institute of Technology, 2009. - Jürgen Kampf, Günter Last, and Ilya Molchanov,
*On the convex hull of symmetric stable processes*, Proc. Amer. Math. Soc.**140**(2012), no. 7, 2527–2535. MR**2898714**, DOI 10.1090/S0002-9939-2012-11128-1 - Daniel A. Klain and Gian-Carlo Rota,
*Introduction to geometric probability*, Lezioni Lincee. [Lincei Lectures], Cambridge University Press, Cambridge, 1997. MR**1608265** - A. N. Kolmogoroff,
*Wienersche Spiralen und einige andere interessante Kurven im Hilbertschen Raum*, C. R. (Doklady) Acad. Sci. URSS (N.S.)**26**(1940), 115–118 (German). MR**0003441** - Hui-Hsiung Kuo,
*Introduction to stochastic integration*, Universitext, Springer, New York, 2006. MR**2180429** - Satya N. Majumdar, Alain Comtet, and Julien Randon-Furling,
*Random convex hulls and extreme value statistics*, J. Stat. Phys.**138**(2010), no. 6, 955–1009. MR**2601420**, DOI 10.1007/s10955-009-9905-z - Jim Pitman and Marc Yor,
*Infinitely divisible laws associated with hyperbolic functions*, Canad. J. Math.**55**(2003), no. 2, 292–330. MR**1969794**, DOI 10.4153/CJM-2003-014-x - Julien Randon-Furling, Satya N. Majumdar, and Alain Comtet,
*Convex hull of $N$ planar Brownian motions: exact results and an application to ecology*, Phys. Rev. Lett.**103**(2009), no. 14, 140602, 4. MR**2551685**, DOI 10.1103/PhysRevLett.103.140602 - Igor Rivin,
*Surface area and other measures of ellipsoids*, Adv. in Appl. Math.**39**(2007), no. 4, 409–427. MR**2356429**, DOI 10.1016/j.aam.2006.08.009 - Rolf Schneider and Wolfgang Weil,
*Stochastic and integral geometry*, Probability and its Applications (New York), Springer-Verlag, Berlin, 2008. MR**2455326**, DOI 10.1007/978-3-540-78859-1 - Scott Sheffield,
*Gaussian free fields for mathematicians*, Probab. Theory Related Fields**139**(2007), no. 3-4, 521–541. MR**2322706**, DOI 10.1007/s00440-006-0050-1 - F. Spitzer and H. Widom,
*The circumference of a convex polygon*, Proc. Amer. Math. Soc.**12**(1961), 506–509. MR**130616**, DOI 10.1090/S0002-9939-1961-0130616-7 - V. N. Sudakov,
*Geometric problems of the theory of infinite-dimensional probability distributions*, Trudy Mat. Inst. Steklov.**141**(1976), 191 (Russian). MR**0431359** - B. S. Tsirel′son,
*A geometric approach to maximum likelihood estimation for an infinite-dimensional Gaussian location. I*, Teor. Veroyatnost. i Primenen.**27**(1982), no. 2, 388–395 (Russian, with English summary). MR**657940** - B. S. Tsirelson,
*A geometric approach to maximum likelihood estimation for an infinite-dimensional Gaussian location. II*, Teor. Veroyatnost. i Primenen.**30**(1985), no. 4, 772–779 (Russian). MR**816291**

## 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
- MR Author ID: 696619
- ORCID: 0000-0001-8483-3373
- 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
- MR Author ID: 744268
- Email: zap1979@gmail.com
- 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
- © Copyright 2016 American Mathematical Society
- Journal: Trans. Amer. Math. Soc.
**368**(2016), 8873-8899 - MSC (2010): Primary 60D05; Secondary 60G15, 52A22
- DOI: https://doi.org/10.1090/tran/6628
- MathSciNet review: 3551592