---
_id: '74'
abstract:
- lang: eng
  text: "We study the Gromov waist in the sense of t-neighborhoods for measures in
    the Euclidean  space,  motivated  by  the  famous  theorem  of  Gromov  about
    \ the  waist  of  radially symmetric Gaussian measures.  In particular, it turns
    our possible to extend Gromov’s original result  to  the  case  of  not  necessarily
    \ radially  symmetric  Gaussian  measure.   We  also  provide examples of measures
    having no t-neighborhood waist property, including a rather wide class\r\nof compactly
    supported radially symmetric measures and their maps into the Euclidean space
    of dimension at least 2.\r\nWe  use  a  simpler  form  of  Gromov’s  pancake  argument
    \ to  produce  some  estimates  of t-neighborhoods of (weighted) volume-critical
    submanifolds in the spirit of the waist theorems, including neighborhoods of algebraic
    manifolds in the complex projective space. In the appendix of this paper we provide
    for reader’s convenience a more detailed explanation of the Caffarelli theorem
    that we use to handle not necessarily radially symmetric Gaussian\r\nmeasures."
article_processing_charge: No
arxiv: 1
author:
- first_name: Arseniy
  full_name: Akopyan, Arseniy
  id: 430D2C90-F248-11E8-B48F-1D18A9856A87
  last_name: Akopyan
  orcid: 0000-0002-2548-617X
- first_name: Roman
  full_name: Karasev, Roman
  last_name: Karasev
citation:
  ama: 'Akopyan A, Karasev R. Gromov’s waist of non-radial Gaussian measures and radial
    non-Gaussian measures. In: Klartag B, Milman E, eds. <i>Geometric Aspects of Functional
    Analysis</i>. Vol 2256. LNM. Springer Nature; 2020:1-27. doi:<a href="https://doi.org/10.1007/978-3-030-36020-7_1">10.1007/978-3-030-36020-7_1</a>'
  apa: Akopyan, A., &#38; Karasev, R. (2020). Gromov’s waist of non-radial Gaussian
    measures and radial non-Gaussian measures. In B. Klartag &#38; E. Milman (Eds.),
    <i>Geometric Aspects of Functional Analysis</i> (Vol. 2256, pp. 1–27). Springer
    Nature. <a href="https://doi.org/10.1007/978-3-030-36020-7_1">https://doi.org/10.1007/978-3-030-36020-7_1</a>
  chicago: Akopyan, Arseniy, and Roman Karasev. “Gromov’s Waist of Non-Radial Gaussian
    Measures and Radial Non-Gaussian Measures.” In <i>Geometric Aspects of Functional
    Analysis</i>, edited by Bo’az Klartag and Emanuel Milman, 2256:1–27. LNM. Springer
    Nature, 2020. <a href="https://doi.org/10.1007/978-3-030-36020-7_1">https://doi.org/10.1007/978-3-030-36020-7_1</a>.
  ieee: A. Akopyan and R. Karasev, “Gromov’s waist of non-radial Gaussian measures
    and radial non-Gaussian measures,” in <i>Geometric Aspects of Functional Analysis</i>,
    vol. 2256, B. Klartag and E. Milman, Eds. Springer Nature, 2020, pp. 1–27.
  ista: 'Akopyan A, Karasev R. 2020.Gromov’s waist of non-radial Gaussian measures
    and radial non-Gaussian measures. In: Geometric Aspects of Functional Analysis.
    vol. 2256, 1–27.'
  mla: Akopyan, Arseniy, and Roman Karasev. “Gromov’s Waist of Non-Radial Gaussian
    Measures and Radial Non-Gaussian Measures.” <i>Geometric Aspects of Functional
    Analysis</i>, edited by Bo’az Klartag and Emanuel Milman, vol. 2256, Springer
    Nature, 2020, pp. 1–27, doi:<a href="https://doi.org/10.1007/978-3-030-36020-7_1">10.1007/978-3-030-36020-7_1</a>.
  short: A. Akopyan, R. Karasev, in:, B. Klartag, E. Milman (Eds.), Geometric Aspects
    of Functional Analysis, Springer Nature, 2020, pp. 1–27.
date_created: 2018-12-11T11:44:29Z
date_published: 2020-06-21T00:00:00Z
date_updated: 2023-08-17T13:48:31Z
day: '21'
department:
- _id: HeEd
- _id: JaMa
doi: 10.1007/978-3-030-36020-7_1
ec_funded: 1
editor:
- first_name: Bo'az
  full_name: Klartag, Bo'az
  last_name: Klartag
- first_name: Emanuel
  full_name: Milman, Emanuel
  last_name: Milman
external_id:
  arxiv:
  - '1808.07350'
  isi:
  - '000557689300003'
intvolume: '      2256'
isi: 1
language:
- iso: eng
main_file_link:
- open_access: '1'
  url: https://arxiv.org/abs/1808.07350
month: '06'
oa: 1
oa_version: Preprint
page: 1-27
project:
- _id: 256E75B8-B435-11E9-9278-68D0E5697425
  call_identifier: H2020
  grant_number: '716117'
  name: Optimal Transport and Stochastic Dynamics
publication: Geometric Aspects of Functional Analysis
publication_identifier:
  eisbn:
  - '9783030360207'
  eissn:
  - '16179692'
  isbn:
  - '9783030360191'
  issn:
  - '00758434'
publication_status: published
publisher: Springer Nature
quality_controlled: '1'
scopus_import: '1'
series_title: LNM
status: public
title: Gromov's waist of non-radial Gaussian measures and radial non-Gaussian measures
type: book_chapter
user_id: 4359f0d1-fa6c-11eb-b949-802e58b17ae8
volume: 2256
year: '2020'
...