---
_id: '9548'
abstract:
- lang: eng
  text: 'We extend the notion of the minimal volume ellipsoid containing a convex
    body in Rd to the setting of logarithmically concave functions. We consider a
    vast class of logarithmically concave functions whose superlevel sets are concentric
    ellipsoids. For a fixed function from this class, we consider the set of all its
    “affine” positions. For any log-concave function f on Rd, we consider functions
    belonging to this set of “affine” positions, and find the one with the minimal
    integral under the condition that it is pointwise greater than or equal to f.
    We study the properties of existence and uniqueness of the solution to this problem.
    For any s∈[0,+∞), we consider the construction dual to the recently defined John
    s-function (Ivanov and Naszódi in Functional John ellipsoids. arXiv preprint:
    arXiv:2006.09934, 2020). We prove that such a construction determines a unique
    function and call it the Löwner s-function of f. We study the Löwner s-functions
    as s tends to zero and to infinity. Finally, extending the notion of the outer
    volume ratio, we define the outer integral ratio of a log-concave function and
    give an asymptotically tight bound on it.'
acknowledgement: The authors acknowledge the support of the grant of the Russian Government
  N 075-15-2019-1926.
article_processing_charge: No
article_type: original
arxiv: 1
author:
- first_name: Grigory
  full_name: Ivanov, Grigory
  id: 87744F66-5C6F-11EA-AFE0-D16B3DDC885E
  last_name: Ivanov
- first_name: Igor
  full_name: Tsiutsiurupa, Igor
  last_name: Tsiutsiurupa
citation:
  ama: Ivanov G, Tsiutsiurupa I. Functional Löwner ellipsoids. <i>Journal of Geometric
    Analysis</i>. 2021;31:11493-11528. doi:<a href="https://doi.org/10.1007/s12220-021-00691-4">10.1007/s12220-021-00691-4</a>
  apa: Ivanov, G., &#38; Tsiutsiurupa, I. (2021). Functional Löwner ellipsoids. <i>Journal
    of Geometric Analysis</i>. Springer. <a href="https://doi.org/10.1007/s12220-021-00691-4">https://doi.org/10.1007/s12220-021-00691-4</a>
  chicago: Ivanov, Grigory, and Igor Tsiutsiurupa. “Functional Löwner Ellipsoids.”
    <i>Journal of Geometric Analysis</i>. Springer, 2021. <a href="https://doi.org/10.1007/s12220-021-00691-4">https://doi.org/10.1007/s12220-021-00691-4</a>.
  ieee: G. Ivanov and I. Tsiutsiurupa, “Functional Löwner ellipsoids,” <i>Journal
    of Geometric Analysis</i>, vol. 31. Springer, pp. 11493–11528, 2021.
  ista: Ivanov G, Tsiutsiurupa I. 2021. Functional Löwner ellipsoids. Journal of Geometric
    Analysis. 31, 11493–11528.
  mla: Ivanov, Grigory, and Igor Tsiutsiurupa. “Functional Löwner Ellipsoids.” <i>Journal
    of Geometric Analysis</i>, vol. 31, Springer, 2021, pp. 11493–528, doi:<a href="https://doi.org/10.1007/s12220-021-00691-4">10.1007/s12220-021-00691-4</a>.
  short: G. Ivanov, I. Tsiutsiurupa, Journal of Geometric Analysis 31 (2021) 11493–11528.
date_created: 2021-06-13T22:01:32Z
date_published: 2021-05-31T00:00:00Z
date_updated: 2023-08-08T14:04:49Z
day: '31'
department:
- _id: UlWa
doi: 10.1007/s12220-021-00691-4
external_id:
  arxiv:
  - '2008.09543'
  isi:
  - '000656507500001'
intvolume: '        31'
isi: 1
language:
- iso: eng
main_file_link:
- open_access: '1'
  url: https://arxiv.org/abs/2008.09543
month: '05'
oa: 1
oa_version: Preprint
page: 11493-11528
publication: Journal of Geometric Analysis
publication_identifier:
  eissn:
  - 1559-002X
  issn:
  - 1050-6926
publication_status: published
publisher: Springer
quality_controlled: '1'
scopus_import: '1'
status: public
title: Functional Löwner ellipsoids
type: journal_article
user_id: 4359f0d1-fa6c-11eb-b949-802e58b17ae8
volume: 31
year: '2021'
...