---
_id: '14660'
abstract:
- lang: eng
text: "The classical Steinitz theorem states that if the origin belongs to the interior
of the convex hull of a set \U0001D446⊂ℝ\U0001D451, then there are at most 2\U0001D451
points of \U0001D446 whose convex hull contains the origin in the interior. Bárány,
Katchalski,and Pach proved the following quantitative version of Steinitz’s theorem.
Let \U0001D444 be a convex polytope in ℝ\U0001D451 containing the standard Euclidean
unit ball \U0001D401\U0001D451. Then there exist at most 2\U0001D451 vertices
of \U0001D444 whose convex hull \U0001D444′ satisfies \U0001D45F\U0001D401\U0001D451⊂\U0001D444′
with \U0001D45F⩾\U0001D451−2\U0001D451. They conjectured that \U0001D45F⩾\U0001D450\U0001D451−1∕2
holds with a universal constant \U0001D450>0. We prove \U0001D45F⩾15\U0001D4512,
the first polynomial lower bound on \U0001D45F. Furthermore, we show that \U0001D45F
is not greater than 2/√\U0001D451."
acknowledgement: M.N. was supported by the János Bolyai Scholarship of the Hungarian
Academy of Sciences aswell as the National Research, Development and Innovation
Fund (NRDI) grants K119670 andK131529, and the ÚNKP-22-5 New National Excellence
Program of the Ministry for Innovationand Technology from the source of the NRDI
as well as the ELTE TKP 2021-NKTA-62 fundingscheme
article_processing_charge: Yes (via OA deal)
article_type: original
arxiv: 1
author:
- first_name: Grigory
full_name: Ivanov, Grigory
id: 87744F66-5C6F-11EA-AFE0-D16B3DDC885E
last_name: Ivanov
- first_name: Márton
full_name: Naszódi, Márton
last_name: Naszódi
citation:
ama: 'Ivanov G, Naszódi M. Quantitative Steinitz theorem: A polynomial bound. Bulletin
of the London Mathematical Society. 2023. doi:10.1112/blms.12965'
apa: 'Ivanov, G., & Naszódi, M. (2023). Quantitative Steinitz theorem: A polynomial
bound. Bulletin of the London Mathematical Society. London Mathematical
Society. https://doi.org/10.1112/blms.12965'
chicago: 'Ivanov, Grigory, and Márton Naszódi. “Quantitative Steinitz Theorem: A
Polynomial Bound.” Bulletin of the London Mathematical Society. London
Mathematical Society, 2023. https://doi.org/10.1112/blms.12965.'
ieee: 'G. Ivanov and M. Naszódi, “Quantitative Steinitz theorem: A polynomial bound,”
Bulletin of the London Mathematical Society. London Mathematical Society,
2023.'
ista: 'Ivanov G, Naszódi M. 2023. Quantitative Steinitz theorem: A polynomial bound.
Bulletin of the London Mathematical Society.'
mla: 'Ivanov, Grigory, and Márton Naszódi. “Quantitative Steinitz Theorem: A Polynomial
Bound.” Bulletin of the London Mathematical Society, London Mathematical
Society, 2023, doi:10.1112/blms.12965.'
short: G. Ivanov, M. Naszódi, Bulletin of the London Mathematical Society (2023).
date_created: 2023-12-10T23:00:58Z
date_published: 2023-12-04T00:00:00Z
date_updated: 2023-12-11T10:03:54Z
day: '04'
department:
- _id: UlWa
doi: 10.1112/blms.12965
external_id:
arxiv:
- '2212.04308'
language:
- iso: eng
main_file_link:
- open_access: '1'
url: ' https://doi.org/10.1112/blms.12965'
month: '12'
oa: 1
oa_version: Published Version
publication: Bulletin of the London Mathematical Society
publication_identifier:
eissn:
- 1469-2120
issn:
- 0024-6093
publication_status: epub_ahead
publisher: London Mathematical Society
quality_controlled: '1'
scopus_import: '1'
status: public
title: 'Quantitative Steinitz theorem: A polynomial bound'
type: journal_article
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
year: '2023'
...
---
_id: '14737'
abstract:
- lang: eng
text: 'John’s fundamental theorem characterizing the largest volume ellipsoid contained
in a convex body $K$ in $\mathbb{R}^{d}$ has seen several generalizations and
extensions. One direction, initiated by V. Milman is to replace ellipsoids by
positions (affine images) of another body $L$. Another, more recent direction
is to consider logarithmically concave functions on $\mathbb{R}^{d}$ instead of
convex bodies: we designate some special, radially symmetric log-concave function
$g$ as the analogue of the Euclidean ball, and want to find its largest integral
position under the constraint that it is pointwise below some given log-concave
function $f$. We follow both directions simultaneously: we consider the functional
question, and allow essentially any meaningful function to play the role of $g$
above. Our general theorems jointly extend known results in both directions. The
dual problem in the setting of convex bodies asks for the smallest volume ellipsoid,
called Löwner’s ellipsoid, containing $K$. We consider the analogous problem for
functions: we characterize the solutions of the optimization problem of finding
a smallest integral position of some log-concave function $g$ under the constraint
that it is pointwise above $f$. It turns out that in the functional setting, the
relationship between the John and the Löwner problems is more intricate than it
is in the setting of convex bodies.'
acknowledgement: "We thank Alexander Litvak for the many discussions on Theorem 1.1.
Igor Tsiutsiurupa participated in the early stage of this project. To our deep regret,
Igor chose another road for his life and stopped working with us.\r\nThis work was
supported by the János Bolyai Scholarship of the Hungarian Academy of Sciences [to
M.N.]; the National Research, Development, and Innovation Fund (NRDI) [K119670 and
K131529 to M.N.]; and the ÚNKP-22-5 New National Excellence Program of the Ministry
for Innovation and Technology from the source of the NRDI [to M.N.]."
article_processing_charge: Yes (via OA deal)
article_type: original
arxiv: 1
author:
- first_name: Grigory
full_name: Ivanov, Grigory
id: 87744F66-5C6F-11EA-AFE0-D16B3DDC885E
last_name: Ivanov
- first_name: Márton
full_name: Naszódi, Márton
last_name: Naszódi
citation:
ama: Ivanov G, Naszódi M. Functional John and Löwner conditions for pairs of log-concave
functions. International Mathematics Research Notices. 2023;2023(23):20613-20669.
doi:10.1093/imrn/rnad210
apa: Ivanov, G., & Naszódi, M. (2023). Functional John and Löwner conditions
for pairs of log-concave functions. International Mathematics Research Notices.
Oxford University Press. https://doi.org/10.1093/imrn/rnad210
chicago: Ivanov, Grigory, and Márton Naszódi. “Functional John and Löwner Conditions
for Pairs of Log-Concave Functions.” International Mathematics Research Notices.
Oxford University Press, 2023. https://doi.org/10.1093/imrn/rnad210.
ieee: G. Ivanov and M. Naszódi, “Functional John and Löwner conditions for pairs
of log-concave functions,” International Mathematics Research Notices,
vol. 2023, no. 23. Oxford University Press, pp. 20613–20669, 2023.
ista: Ivanov G, Naszódi M. 2023. Functional John and Löwner conditions for pairs
of log-concave functions. International Mathematics Research Notices. 2023(23),
20613–20669.
mla: Ivanov, Grigory, and Márton Naszódi. “Functional John and Löwner Conditions
for Pairs of Log-Concave Functions.” International Mathematics Research Notices,
vol. 2023, no. 23, Oxford University Press, 2023, pp. 20613–69, doi:10.1093/imrn/rnad210.
short: G. Ivanov, M. Naszódi, International Mathematics Research Notices 2023 (2023)
20613–20669.
date_created: 2024-01-08T09:48:56Z
date_published: 2023-12-01T00:00:00Z
date_updated: 2024-01-08T09:57:25Z
day: '01'
ddc:
- '510'
department:
- _id: UlWa
doi: 10.1093/imrn/rnad210
external_id:
arxiv:
- '2212.11781'
file:
- access_level: open_access
checksum: 353666cea80633beb0f1ffd342dff6d4
content_type: application/pdf
creator: dernst
date_created: 2024-01-08T09:53:09Z
date_updated: 2024-01-08T09:53:09Z
file_id: '14738'
file_name: 2023_IMRN_Ivanov.pdf
file_size: 815777
relation: main_file
success: 1
file_date_updated: 2024-01-08T09:53:09Z
has_accepted_license: '1'
intvolume: ' 2023'
issue: '23'
keyword:
- General Mathematics
language:
- iso: eng
license: https://creativecommons.org/licenses/by-nc-nd/4.0/
month: '12'
oa: 1
oa_version: Published Version
page: 20613-20669
publication: International Mathematics Research Notices
publication_identifier:
eissn:
- 1687-0247
issn:
- 1073-7928
publication_status: published
publisher: Oxford University Press
quality_controlled: '1'
status: public
title: Functional John and Löwner conditions for pairs of log-concave functions
tmp:
image: /images/cc_by_nc_nd.png
legal_code_url: https://creativecommons.org/licenses/by-nc-nd/4.0/legalcode
name: Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International
(CC BY-NC-ND 4.0)
short: CC BY-NC-ND (4.0)
type: journal_article
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
volume: 2023
year: '2023'
...
---
_id: '12680'
abstract:
- lang: eng
text: The celebrated Erdős–Ko–Rado theorem about the maximal size of an intersecting
family of r-element subsets of was extended to the setting of exterior algebra
in [5, Theorem 2.3] and in [6, Theorem 1.4]. However, the equality case has not
been settled yet. In this short note, we show that the extension of the Erdős–Ko–Rado
theorem and the characterization of the equality case therein, as well as those
of the Hilton–Milner theorem to the setting of exterior algebra in the simplest
non-trivial case of two-forms follow from a folklore puzzle about possible arrangements
of an intersecting family of lines.
article_number: '113363'
article_processing_charge: No
article_type: letter_note
arxiv: 1
author:
- first_name: Grigory
full_name: Ivanov, Grigory
id: 87744F66-5C6F-11EA-AFE0-D16B3DDC885E
last_name: Ivanov
- first_name: Seyda
full_name: Köse, Seyda
id: 8ba3170d-dc85-11ea-9058-c4251c96a6eb
last_name: Köse
citation:
ama: Ivanov G, Köse S. Erdős-Ko-Rado and Hilton-Milner theorems for two-forms. Discrete
Mathematics. 2023;346(6). doi:10.1016/j.disc.2023.113363
apa: Ivanov, G., & Köse, S. (2023). Erdős-Ko-Rado and Hilton-Milner theorems
for two-forms. Discrete Mathematics. Elsevier. https://doi.org/10.1016/j.disc.2023.113363
chicago: Ivanov, Grigory, and Seyda Köse. “Erdős-Ko-Rado and Hilton-Milner Theorems
for Two-Forms.” Discrete Mathematics. Elsevier, 2023. https://doi.org/10.1016/j.disc.2023.113363.
ieee: G. Ivanov and S. Köse, “Erdős-Ko-Rado and Hilton-Milner theorems for two-forms,”
Discrete Mathematics, vol. 346, no. 6. Elsevier, 2023.
ista: Ivanov G, Köse S. 2023. Erdős-Ko-Rado and Hilton-Milner theorems for two-forms.
Discrete Mathematics. 346(6), 113363.
mla: Ivanov, Grigory, and Seyda Köse. “Erdős-Ko-Rado and Hilton-Milner Theorems
for Two-Forms.” Discrete Mathematics, vol. 346, no. 6, 113363, Elsevier,
2023, doi:10.1016/j.disc.2023.113363.
short: G. Ivanov, S. Köse, Discrete Mathematics 346 (2023).
date_created: 2023-02-26T23:01:00Z
date_published: 2023-06-01T00:00:00Z
date_updated: 2023-10-04T11:54:57Z
day: '01'
department:
- _id: UlWa
- _id: GradSch
doi: 10.1016/j.disc.2023.113363
external_id:
arxiv:
- '2201.10892'
intvolume: ' 346'
issue: '6'
language:
- iso: eng
main_file_link:
- open_access: '1'
url: ' https://doi.org/10.48550/arXiv.2201.10892'
month: '06'
oa: 1
oa_version: Preprint
publication: Discrete Mathematics
publication_identifier:
issn:
- 0012-365X
publication_status: published
publisher: Elsevier
quality_controlled: '1'
related_material:
record:
- id: '13331'
relation: dissertation_contains
status: public
scopus_import: '1'
status: public
title: Erdős-Ko-Rado and Hilton-Milner theorems for two-forms
type: journal_article
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
volume: 346
year: '2023'
...
---
_id: '10887'
abstract:
- lang: eng
text: "We introduce a new way of representing logarithmically concave functions
on Rd. It allows us to extend the notion of the largest volume ellipsoid contained
in a convex body to the setting of logarithmically concave functions as follows.
For every s>0, we define a class of non-negative functions on Rd derived from
ellipsoids in Rd+1. For any log-concave function f on Rd , and any fixed s>0,
we consider functions belonging to this class, and find the one with the largest
integral under the condition that it is pointwise less than or equal to f, and
we call it the John s-function of f. After establishing existence and uniqueness,
we give a characterization of this function similar to the one given by John in
his fundamental theorem. We find that John s-functions converge to characteristic
functions of ellipsoids as s tends to zero and to Gaussian densities as s tends
to infinity.\r\nAs an application, we prove a quantitative Helly type result:
the integral of the pointwise minimum of any family of log-concave functions is
at least a constant cd multiple of the integral of the pointwise minimum of a
properly chosen subfamily of size 3d+2, where cd depends only on d."
acknowledgement: 'G.I. was supported by the Ministry of Education and Science of the
Russian Federation in the framework of MegaGrant no 075-15-2019-1926. M.N. was supported
by the National Research, Development and Innovation Fund (NRDI) grants K119670
and KKP-133864 as well as the Bolyai Scholarship of the Hungarian Academy of Sciences
and the New National Excellence Programme and the TKP2020-NKA-06 program provided
by the NRDI. '
article_number: '109441'
article_processing_charge: Yes (via OA deal)
article_type: original
arxiv: 1
author:
- first_name: Grigory
full_name: Ivanov, Grigory
id: 87744F66-5C6F-11EA-AFE0-D16B3DDC885E
last_name: Ivanov
- first_name: Márton
full_name: Naszódi, Márton
last_name: Naszódi
citation:
ama: Ivanov G, Naszódi M. Functional John ellipsoids. Journal of Functional Analysis.
2022;282(11). doi:10.1016/j.jfa.2022.109441
apa: Ivanov, G., & Naszódi, M. (2022). Functional John ellipsoids. Journal
of Functional Analysis. Elsevier. https://doi.org/10.1016/j.jfa.2022.109441
chicago: Ivanov, Grigory, and Márton Naszódi. “Functional John Ellipsoids.” Journal
of Functional Analysis. Elsevier, 2022. https://doi.org/10.1016/j.jfa.2022.109441.
ieee: G. Ivanov and M. Naszódi, “Functional John ellipsoids,” Journal of Functional
Analysis, vol. 282, no. 11. Elsevier, 2022.
ista: Ivanov G, Naszódi M. 2022. Functional John ellipsoids. Journal of Functional
Analysis. 282(11), 109441.
mla: Ivanov, Grigory, and Márton Naszódi. “Functional John Ellipsoids.” Journal
of Functional Analysis, vol. 282, no. 11, 109441, Elsevier, 2022, doi:10.1016/j.jfa.2022.109441.
short: G. Ivanov, M. Naszódi, Journal of Functional Analysis 282 (2022).
date_created: 2022-03-20T23:01:38Z
date_published: 2022-06-01T00:00:00Z
date_updated: 2023-08-02T14:51:11Z
day: '01'
ddc:
- '510'
department:
- _id: UlWa
doi: 10.1016/j.jfa.2022.109441
external_id:
arxiv:
- '2006.09934'
isi:
- '000781371300008'
file:
- access_level: open_access
checksum: 1cf185e264e04c87cb1ce67a00db88ab
content_type: application/pdf
creator: dernst
date_created: 2022-08-02T10:40:48Z
date_updated: 2022-08-02T10:40:48Z
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file_name: 2022_JourFunctionalAnalysis_Ivanov.pdf
file_size: 734482
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intvolume: ' 282'
isi: 1
issue: '11'
language:
- iso: eng
month: '06'
oa: 1
oa_version: Published Version
publication: Journal of Functional Analysis
publication_identifier:
eissn:
- 1096-0783
issn:
- 0022-1236
publication_status: published
publisher: Elsevier
quality_controlled: '1'
scopus_import: '1'
status: public
title: Functional John ellipsoids
tmp:
image: /images/cc_by.png
legal_code_url: https://creativecommons.org/licenses/by/4.0/legalcode
name: Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)
short: CC BY (4.0)
type: journal_article
user_id: 4359f0d1-fa6c-11eb-b949-802e58b17ae8
volume: 282
year: '2022'
...
---
_id: '11435'
abstract:
- lang: eng
text: 'We introduce a new variant of quantitative Helly-type theorems: the minimal
homothetic distance of the intersection of a family of convex sets to the intersection
of a subfamily of a fixed size. As an application, we establish the following
quantitative Helly-type result for the diameter. If $K$ is the intersection of
finitely many convex bodies in $\mathbb{R}^d$, then one can select $2d$ of these
bodies whose intersection is of diameter at most $(2d)^3{diam}(K)$. The best previously
known estimate, due to Brazitikos [Bull. Hellenic Math. Soc., 62 (2018), pp. 19--25],
is $c d^{11/2}$. Moreover, we confirm that the multiplicative factor $c d^{1/2}$
conjectured by Bárány, Katchalski, and Pach [Proc. Amer. Math. Soc., 86 (1982),
pp. 109--114] cannot be improved. The bounds above follow from our key result
that concerns sparse approximation of a convex polytope by the convex hull of
a well-chosen subset of its vertices: Assume that $Q \subset {\mathbb R}^d$ is
a polytope whose centroid is the origin. Then there exist at most 2d vertices
of $Q$ whose convex hull $Q^{\prime \prime}$ satisfies $Q \subset - 8d^3 Q^{\prime
\prime}.$'
acknowledgement: "G.I. acknowledges the financial support from the Ministry of Educational
and Science of the Russian Federation in the framework of MegaGrant no 075-15-2019-1926.
M.N. was supported by the National Research, Development and Innovation Fund (NRDI)
grants K119670 and\r\nKKP-133864 as well as the Bolyai Scholarship of the Hungarian
Academy of Sciences and the New National Excellence Programme and the TKP2020-NKA-06
program provided by the NRDI."
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: Marton
full_name: Naszodi, Marton
last_name: Naszodi
citation:
ama: 'Ivanov G, Naszodi M. A quantitative Helly-type theorem: Containment in a homothet.
SIAM Journal on Discrete Mathematics. 2022;36(2):951-957. doi:10.1137/21M1403308'
apa: 'Ivanov, G., & Naszodi, M. (2022). A quantitative Helly-type theorem: Containment
in a homothet. SIAM Journal on Discrete Mathematics. Society for Industrial
and Applied Mathematics. https://doi.org/10.1137/21M1403308'
chicago: 'Ivanov, Grigory, and Marton Naszodi. “A Quantitative Helly-Type Theorem:
Containment in a Homothet.” SIAM Journal on Discrete Mathematics. Society
for Industrial and Applied Mathematics, 2022. https://doi.org/10.1137/21M1403308.'
ieee: 'G. Ivanov and M. Naszodi, “A quantitative Helly-type theorem: Containment
in a homothet,” SIAM Journal on Discrete Mathematics, vol. 36, no. 2. Society
for Industrial and Applied Mathematics, pp. 951–957, 2022.'
ista: 'Ivanov G, Naszodi M. 2022. A quantitative Helly-type theorem: Containment
in a homothet. SIAM Journal on Discrete Mathematics. 36(2), 951–957.'
mla: 'Ivanov, Grigory, and Marton Naszodi. “A Quantitative Helly-Type Theorem: Containment
in a Homothet.” SIAM Journal on Discrete Mathematics, vol. 36, no. 2, Society
for Industrial and Applied Mathematics, 2022, pp. 951–57, doi:10.1137/21M1403308.'
short: G. Ivanov, M. Naszodi, SIAM Journal on Discrete Mathematics 36 (2022) 951–957.
date_created: 2022-06-05T22:01:50Z
date_published: 2022-04-11T00:00:00Z
date_updated: 2023-10-18T06:58:03Z
day: '11'
department:
- _id: UlWa
doi: 10.1137/21M1403308
external_id:
arxiv:
- '2103.04122'
isi:
- '000793158200002'
intvolume: ' 36'
isi: 1
issue: '2'
language:
- iso: eng
main_file_link:
- open_access: '1'
url: ' https://doi.org/10.48550/arXiv.2103.04122'
month: '04'
oa: 1
oa_version: Preprint
page: 951-957
publication: SIAM Journal on Discrete Mathematics
publication_identifier:
issn:
- 0895-4801
publication_status: published
publisher: Society for Industrial and Applied Mathematics
quality_controlled: '1'
scopus_import: '1'
status: public
title: 'A quantitative Helly-type theorem: Containment in a homothet'
type: journal_article
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
volume: 36
year: '2022'
...
---
_id: '10856'
abstract:
- lang: eng
text: "We study the properties of the maximal volume k-dimensional sections of the
n-dimensional cube [−1, 1]n. We obtain a first order necessary condition for a
k-dimensional subspace to be a local maximizer of the volume of such sections,
which we formulate in a geometric way. We estimate the length of the projection
of a vector of the standard basis of Rn onto a k-dimensional subspace that maximizes
the volume of the intersection. We \x1Cnd the optimal upper bound on the volume
of a planar section of the cube [−1, 1]n , n ≥ 2."
acknowledgement: "The authors acknowledge the support of the grant of the Russian
Government N 075-15-\r\n2019-1926. G.I.was supported also by the SwissNational Science
Foundation grant 200021-179133. The authors are very grateful to the anonymous reviewer
for valuable remarks."
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. On the volume of sections of the cube. Analysis
and Geometry in Metric Spaces. 2021;9(1):1-18. doi:10.1515/agms-2020-0103
apa: Ivanov, G., & Tsiutsiurupa, I. (2021). On the volume of sections of the
cube. Analysis and Geometry in Metric Spaces. De Gruyter. https://doi.org/10.1515/agms-2020-0103
chicago: Ivanov, Grigory, and Igor Tsiutsiurupa. “On the Volume of Sections of the
Cube.” Analysis and Geometry in Metric Spaces. De Gruyter, 2021. https://doi.org/10.1515/agms-2020-0103.
ieee: G. Ivanov and I. Tsiutsiurupa, “On the volume of sections of the cube,” Analysis
and Geometry in Metric Spaces, vol. 9, no. 1. De Gruyter, pp. 1–18, 2021.
ista: Ivanov G, Tsiutsiurupa I. 2021. On the volume of sections of the cube. Analysis
and Geometry in Metric Spaces. 9(1), 1–18.
mla: Ivanov, Grigory, and Igor Tsiutsiurupa. “On the Volume of Sections of the Cube.”
Analysis and Geometry in Metric Spaces, vol. 9, no. 1, De Gruyter, 2021,
pp. 1–18, doi:10.1515/agms-2020-0103.
short: G. Ivanov, I. Tsiutsiurupa, Analysis and Geometry in Metric Spaces 9 (2021)
1–18.
date_created: 2022-03-18T09:25:14Z
date_published: 2021-01-29T00:00:00Z
date_updated: 2023-08-17T07:07:58Z
day: '29'
ddc:
- '510'
department:
- _id: UlWa
doi: 10.1515/agms-2020-0103
external_id:
arxiv:
- '2004.02674'
isi:
- '000734286800001'
file:
- access_level: open_access
checksum: 7e615ac8489f5eae580b6517debfdc53
content_type: application/pdf
creator: dernst
date_created: 2022-03-18T09:31:59Z
date_updated: 2022-03-18T09:31:59Z
file_id: '10857'
file_name: 2021_AnalysisMetricSpaces_Ivanov.pdf
file_size: 789801
relation: main_file
success: 1
file_date_updated: 2022-03-18T09:31:59Z
has_accepted_license: '1'
intvolume: ' 9'
isi: 1
issue: '1'
keyword:
- Applied Mathematics
- Geometry and Topology
- Analysis
language:
- iso: eng
month: '01'
oa: 1
oa_version: Published Version
page: 1-18
publication: Analysis and Geometry in Metric Spaces
publication_identifier:
issn:
- 2299-3274
publication_status: published
publisher: De Gruyter
quality_controlled: '1'
scopus_import: '1'
status: public
title: On the volume of sections of the cube
tmp:
image: /images/cc_by.png
legal_code_url: https://creativecommons.org/licenses/by/4.0/legalcode
name: Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)
short: CC BY (4.0)
type: journal_article
user_id: 4359f0d1-fa6c-11eb-b949-802e58b17ae8
volume: 9
year: '2021'
...
---
_id: '10860'
abstract:
- lang: eng
text: A tight frame is the orthogonal projection of some orthonormal basis of Rn
onto Rk. We show that a set of vectors is a tight frame if and only if the set
of all cross products of these vectors is a tight frame. We reformulate a range
of problems on the volume of projections (or sections) of regular polytopes in
terms of tight frames and write a first-order necessary condition for local extrema
of these problems. As applications, we prove new results for the problem of maximization
of the volume of zonotopes.
acknowledgement: The author was supported by the Swiss National Science Foundation
grant 200021_179133. The author acknowledges the financial support from the Ministry
of Education and Science of the Russian Federation in the framework of MegaGrant
no. 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
citation:
ama: Ivanov G. Tight frames and related geometric problems. Canadian Mathematical
Bulletin. 2021;64(4):942-963. doi:10.4153/s000843952000096x
apa: Ivanov, G. (2021). Tight frames and related geometric problems. Canadian
Mathematical Bulletin. Canadian Mathematical Society. https://doi.org/10.4153/s000843952000096x
chicago: Ivanov, Grigory. “Tight Frames and Related Geometric Problems.” Canadian
Mathematical Bulletin. Canadian Mathematical Society, 2021. https://doi.org/10.4153/s000843952000096x.
ieee: G. Ivanov, “Tight frames and related geometric problems,” Canadian Mathematical
Bulletin, vol. 64, no. 4. Canadian Mathematical Society, pp. 942–963, 2021.
ista: Ivanov G. 2021. Tight frames and related geometric problems. Canadian Mathematical
Bulletin. 64(4), 942–963.
mla: Ivanov, Grigory. “Tight Frames and Related Geometric Problems.” Canadian
Mathematical Bulletin, vol. 64, no. 4, Canadian Mathematical Society, 2021,
pp. 942–63, doi:10.4153/s000843952000096x.
short: G. Ivanov, Canadian Mathematical Bulletin 64 (2021) 942–963.
date_created: 2022-03-18T09:55:59Z
date_published: 2021-12-18T00:00:00Z
date_updated: 2023-09-05T12:43:09Z
day: '18'
department:
- _id: UlWa
doi: 10.4153/s000843952000096x
external_id:
arxiv:
- '1804.10055'
isi:
- '000730165300021'
intvolume: ' 64'
isi: 1
issue: '4'
keyword:
- General Mathematics
- Tight frame
- Grassmannian
- zonotope
language:
- iso: eng
main_file_link:
- open_access: '1'
url: https://arxiv.org/abs/1804.10055
month: '12'
oa: 1
oa_version: Preprint
page: 942-963
publication: Canadian Mathematical Bulletin
publication_identifier:
eissn:
- 1496-4287
issn:
- 0008-4395
publication_status: published
publisher: Canadian Mathematical Society
quality_controlled: '1'
scopus_import: '1'
status: public
title: Tight frames and related geometric problems
type: journal_article
user_id: c635000d-4b10-11ee-a964-aac5a93f6ac1
volume: 64
year: '2021'
...
---
_id: '9037'
abstract:
- lang: eng
text: "We continue our study of ‘no‐dimension’ analogues of basic theorems in combinatorial
and convex geometry in Banach spaces. We generalize some results of the paper
(Adiprasito, Bárány and Mustafa, ‘Theorems of Carathéodory, Helly, and Tverberg
without dimension’, Proceedings of the Thirtieth Annual ACM‐SIAM Symposium on
Discrete Algorithms (Society for Industrial and Applied Mathematics, San Diego,
California, 2019) 2350–2360) and prove no‐dimension versions of the colored Tverberg
theorem, the selection lemma and the weak \U0001D700 ‐net theorem in Banach spaces
of type \U0001D45D>1 . To prove these results, we use the original ideas of Adiprasito,
Bárány and Mustafa for the Euclidean case, our no‐dimension version of the Radon
theorem and slightly modified version of the celebrated Maurey lemma."
acknowledgement: "I wish to thank Imre Bárány for bringing the problem to my attention.
I am grateful to Marton Naszódi and Igor Tsiutsiurupa for useful remarks and help
with the text.\r\nThe author acknowledges the financial support from the Ministry
of Educational and Science of the Russian Federation in the framework of MegaGrant
no 075‐15‐2019‐1926."
article_processing_charge: Yes (via OA deal)
article_type: original
arxiv: 1
author:
- first_name: Grigory
full_name: Ivanov, Grigory
id: 87744F66-5C6F-11EA-AFE0-D16B3DDC885E
last_name: Ivanov
citation:
ama: Ivanov G. No-dimension Tverberg’s theorem and its corollaries in Banach spaces
of type p. Bulletin of the London Mathematical Society. 2021;53(2):631-641.
doi:10.1112/blms.12449
apa: Ivanov, G. (2021). No-dimension Tverberg’s theorem and its corollaries in Banach
spaces of type p. Bulletin of the London Mathematical Society. London Mathematical
Society. https://doi.org/10.1112/blms.12449
chicago: Ivanov, Grigory. “No-Dimension Tverberg’s Theorem and Its Corollaries in
Banach Spaces of Type P.” Bulletin of the London Mathematical Society.
London Mathematical Society, 2021. https://doi.org/10.1112/blms.12449.
ieee: G. Ivanov, “No-dimension Tverberg’s theorem and its corollaries in Banach
spaces of type p,” Bulletin of the London Mathematical Society, vol. 53,
no. 2. London Mathematical Society, pp. 631–641, 2021.
ista: Ivanov G. 2021. No-dimension Tverberg’s theorem and its corollaries in Banach
spaces of type p. Bulletin of the London Mathematical Society. 53(2), 631–641.
mla: Ivanov, Grigory. “No-Dimension Tverberg’s Theorem and Its Corollaries in Banach
Spaces of Type P.” Bulletin of the London Mathematical Society, vol. 53,
no. 2, London Mathematical Society, 2021, pp. 631–41, doi:10.1112/blms.12449.
short: G. Ivanov, Bulletin of the London Mathematical Society 53 (2021) 631–641.
date_created: 2021-01-24T23:01:08Z
date_published: 2021-04-01T00:00:00Z
date_updated: 2023-08-07T13:35:20Z
day: '01'
ddc:
- '510'
department:
- _id: UlWa
doi: 10.1112/blms.12449
external_id:
arxiv:
- '1912.08561'
isi:
- '000607265100001'
file:
- access_level: open_access
checksum: e6ceaa6470d835eb4c211cbdd38fdfd1
content_type: application/pdf
creator: kschuh
date_created: 2021-08-06T09:59:45Z
date_updated: 2021-08-06T09:59:45Z
file_id: '9796'
file_name: 2021_BLMS_Ivanov.pdf
file_size: 194550
relation: main_file
success: 1
file_date_updated: 2021-08-06T09:59:45Z
has_accepted_license: '1'
intvolume: ' 53'
isi: 1
issue: '2'
language:
- iso: eng
month: '04'
oa: 1
oa_version: Published Version
page: 631-641
publication: Bulletin of the London Mathematical Society
publication_identifier:
eissn:
- '14692120'
issn:
- '00246093'
publication_status: published
publisher: London Mathematical Society
quality_controlled: '1'
scopus_import: '1'
status: public
title: No-dimension Tverberg's theorem and its corollaries in Banach spaces of type
p
tmp:
image: /images/cc_by_nc_nd.png
legal_code_url: https://creativecommons.org/licenses/by-nc-nd/4.0/legalcode
name: Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International
(CC BY-NC-ND 4.0)
short: CC BY-NC-ND (4.0)
type: journal_article
user_id: 4359f0d1-fa6c-11eb-b949-802e58b17ae8
volume: 53
year: '2021'
...
---
_id: '9098'
abstract:
- lang: eng
text: "We study properties of the volume of projections of the n-dimensional\r\ncross-polytope
$\\crosp^n = \\{ x \\in \\R^n \\mid |x_1| + \\dots + |x_n| \\leqslant 1\\}.$ We
prove that the projection of $\\crosp^n$ onto a k-dimensional coordinate subspace
has the maximum possible volume for k=2 and for k=3.\r\nWe obtain the exact lower
bound on the volume of such a projection onto a two-dimensional plane. Also, we
show that there exist local maxima which are not global ones for the volume of
a projection of $\\crosp^n$ onto a k-dimensional subspace for any n>k⩾2."
acknowledgement: Research was supported by the Russian Foundation for Basic Research,
project 18-01-00036A (Theorems 1.5 and 5.3) and by the Ministry of Education and
Science of the Russian Federation in the framework of MegaGrant no 075-15-2019-1926
(Theorems 1.2 and 7.3).
article_number: '112312'
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
citation:
ama: Ivanov G. On the volume of projections of the cross-polytope. Discrete Mathematics.
2021;344(5). doi:10.1016/j.disc.2021.112312
apa: Ivanov, G. (2021). On the volume of projections of the cross-polytope. Discrete
Mathematics. Elsevier. https://doi.org/10.1016/j.disc.2021.112312
chicago: Ivanov, Grigory. “On the Volume of Projections of the Cross-Polytope.”
Discrete Mathematics. Elsevier, 2021. https://doi.org/10.1016/j.disc.2021.112312.
ieee: G. Ivanov, “On the volume of projections of the cross-polytope,” Discrete
Mathematics, vol. 344, no. 5. Elsevier, 2021.
ista: Ivanov G. 2021. On the volume of projections of the cross-polytope. Discrete
Mathematics. 344(5), 112312.
mla: Ivanov, Grigory. “On the Volume of Projections of the Cross-Polytope.” Discrete
Mathematics, vol. 344, no. 5, 112312, Elsevier, 2021, doi:10.1016/j.disc.2021.112312.
short: G. Ivanov, Discrete Mathematics 344 (2021).
date_created: 2021-02-07T23:01:12Z
date_published: 2021-05-01T00:00:00Z
date_updated: 2023-08-07T13:40:37Z
day: '01'
department:
- _id: UlWa
doi: 10.1016/j.disc.2021.112312
external_id:
arxiv:
- '1808.09165'
isi:
- '000633365200001'
intvolume: ' 344'
isi: 1
issue: '5'
language:
- iso: eng
main_file_link:
- open_access: '1'
url: https://arxiv.org/abs/1808.09165
month: '05'
oa: 1
oa_version: Preprint
publication: Discrete Mathematics
publication_identifier:
issn:
- 0012365X
publication_status: published
publisher: Elsevier
quality_controlled: '1'
scopus_import: '1'
status: public
title: On the volume of projections of the cross-polytope
type: journal_article
user_id: 4359f0d1-fa6c-11eb-b949-802e58b17ae8
volume: 344
year: '2021'
...
---
_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. Journal of Geometric
Analysis. 2021;31:11493-11528. doi:10.1007/s12220-021-00691-4
apa: Ivanov, G., & Tsiutsiurupa, I. (2021). Functional Löwner ellipsoids. Journal
of Geometric Analysis. Springer. https://doi.org/10.1007/s12220-021-00691-4
chicago: Ivanov, Grigory, and Igor Tsiutsiurupa. “Functional Löwner Ellipsoids.”
Journal of Geometric Analysis. Springer, 2021. https://doi.org/10.1007/s12220-021-00691-4.
ieee: G. Ivanov and I. Tsiutsiurupa, “Functional Löwner ellipsoids,” Journal
of Geometric Analysis, 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.” Journal
of Geometric Analysis, vol. 31, Springer, 2021, pp. 11493–528, doi:10.1007/s12220-021-00691-4.
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'
...
---
_id: '10181'
abstract:
- lang: eng
text: In this article we study some geometric properties of proximally smooth sets.
First, we introduce a modification of the metric projection and prove its existence.
Then we provide an algorithm for constructing a rectifiable curve between two
sufficiently close points of a proximally smooth set in a uniformly convex and
uniformly smooth Banach space, with the moduli of smoothness and convexity of
power type. Our algorithm returns a reasonably short curve between two sufficiently
close points of a proximally smooth set, is iterative and uses our modification
of the metric projection. We estimate the length of the constructed curve and
its deviation from the segment with the same endpoints. These estimates coincide
up to a constant factor with those for the geodesics in a proximally smooth set
in a Hilbert space.
acknowledgement: Theorem 2 was obtained at Steklov Mathematical Institute RAS and
supported by Russian Science Foundation, grant N 19-11-00087.
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: Mariana S.
full_name: Lopushanski, Mariana S.
last_name: Lopushanski
citation:
ama: Ivanov G, Lopushanski MS. Rectifiable curves in proximally smooth sets. Set-Valued
and Variational Analysis. 2021. doi:10.1007/s11228-021-00612-1
apa: Ivanov, G., & Lopushanski, M. S. (2021). Rectifiable curves in proximally
smooth sets. Set-Valued and Variational Analysis. Springer Nature. https://doi.org/10.1007/s11228-021-00612-1
chicago: Ivanov, Grigory, and Mariana S. Lopushanski. “Rectifiable Curves in Proximally
Smooth Sets.” Set-Valued and Variational Analysis. Springer Nature, 2021.
https://doi.org/10.1007/s11228-021-00612-1.
ieee: G. Ivanov and M. S. Lopushanski, “Rectifiable curves in proximally smooth
sets,” Set-Valued and Variational Analysis. Springer Nature, 2021.
ista: Ivanov G, Lopushanski MS. 2021. Rectifiable curves in proximally smooth sets.
Set-Valued and Variational Analysis.
mla: Ivanov, Grigory, and Mariana S. Lopushanski. “Rectifiable Curves in Proximally
Smooth Sets.” Set-Valued and Variational Analysis, Springer Nature, 2021,
doi:10.1007/s11228-021-00612-1.
short: G. Ivanov, M.S. Lopushanski, Set-Valued and Variational Analysis (2021).
date_created: 2021-10-24T22:01:35Z
date_published: 2021-10-09T00:00:00Z
date_updated: 2023-08-14T08:11:38Z
day: '09'
department:
- _id: UlWa
doi: 10.1007/s11228-021-00612-1
external_id:
arxiv:
- '2012.10691'
isi:
- '000705774800001'
isi: 1
language:
- iso: eng
main_file_link:
- open_access: '1'
url: https://arxiv.org/abs/2012.10691
month: '10'
oa: 1
oa_version: Published Version
publication: Set-Valued and Variational Analysis
publication_identifier:
eissn:
- 1877-0541
issn:
- 0927-6947
publication_status: published
publisher: Springer Nature
quality_controlled: '1'
scopus_import: '1'
status: public
title: Rectifiable curves in proximally smooth sets
type: journal_article
user_id: 4359f0d1-fa6c-11eb-b949-802e58b17ae8
year: '2021'
...