Min-Max Polarization for Certain Classes of Sharp Configurations on the Sphere
We consider the problem of finding an N -point configuration on the sphere S d ⊂ R d + 1 with the smallest absolute maximum value over S d of its total potential. The potential induced by each point y in a given configuration at a point x ∈ S d is f x - y 2 , where f is continuous on [0, 4] and comp...
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| Veröffentlicht in: | Constructive approximation 2024-10, Vol.60 (2), p.237-252 |
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| container_title | Constructive approximation |
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| creator | Borodachov, Sergiy |
| description | We consider the problem of finding an
N
-point configuration on the sphere
S
d
⊂
R
d
+
1
with the smallest absolute maximum value over
S
d
of its total potential. The potential induced by each point
y
in a given configuration at a point
x
∈
S
d
is
f
x
-
y
2
, where
f
is continuous on [0, 4] and completely monotone on (0, 4], and
x
-
y
is the Euclidean distance between points
x
and
y
. We show that any sharp point configuration
ω
¯
N
on
S
d
, which is antipodal or is a spherical design of an even strength is a solution to this problem. We also prove that the absolute maximum over
S
d
of the potential of any such configuration
ω
¯
N
is attained at points of
ω
¯
N
. |
| doi_str_mv | 10.1007/s00365-023-09661-1 |
| format | Article |
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N
-point configuration on the sphere
S
d
⊂
R
d
+
1
with the smallest absolute maximum value over
S
d
of its total potential. The potential induced by each point
y
in a given configuration at a point
x
∈
S
d
is
f
x
-
y
2
, where
f
is continuous on [0, 4] and completely monotone on (0, 4], and
x
-
y
is the Euclidean distance between points
x
and
y
. We show that any sharp point configuration
ω
¯
N
on
S
d
, which is antipodal or is a spherical design of an even strength is a solution to this problem. We also prove that the absolute maximum over
S
d
of the potential of any such configuration
ω
¯
N
is attained at points of
ω
¯
N
.</description><identifier>ISSN: 0176-4276</identifier><identifier>EISSN: 1432-0940</identifier><identifier>DOI: 10.1007/s00365-023-09661-1</identifier><language>eng</language><publisher>New York: Springer US</publisher><subject>Analysis ; Configuration management ; Euclidean geometry ; Mathematics ; Mathematics and Statistics ; Numerical Analysis</subject><ispartof>Constructive approximation, 2024-10, Vol.60 (2), p.237-252</ispartof><rights>The Author(s), under exclusive licence to Springer Science+Business Media, LLC, part of Springer Nature 2023. Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.</rights><lds50>peer_reviewed</lds50><woscitedreferencessubscribed>false</woscitedreferencessubscribed><citedby>FETCH-LOGICAL-c319t-f78b016d241320e0a6ce7289242de5bb641902b8fea264b4b0d234d5f98b03213</citedby><cites>FETCH-LOGICAL-c319t-f78b016d241320e0a6ce7289242de5bb641902b8fea264b4b0d234d5f98b03213</cites></display><links><openurl>$$Topenurl_article</openurl><openurlfulltext>$$Topenurlfull_article</openurlfulltext><thumbnail>$$Tsyndetics_thumb_exl</thumbnail><linktopdf>$$Uhttps://link.springer.com/content/pdf/10.1007/s00365-023-09661-1$$EPDF$$P50$$Gspringer$$H</linktopdf><linktohtml>$$Uhttps://link.springer.com/10.1007/s00365-023-09661-1$$EHTML$$P50$$Gspringer$$H</linktohtml><link.rule.ids>314,776,780,27901,27902,41464,42533,51294</link.rule.ids></links><search><creatorcontrib>Borodachov, Sergiy</creatorcontrib><title>Min-Max Polarization for Certain Classes of Sharp Configurations on the Sphere</title><title>Constructive approximation</title><addtitle>Constr Approx</addtitle><description>We consider the problem of finding an
N
-point configuration on the sphere
S
d
⊂
R
d
+
1
with the smallest absolute maximum value over
S
d
of its total potential. The potential induced by each point
y
in a given configuration at a point
x
∈
S
d
is
f
x
-
y
2
, where
f
is continuous on [0, 4] and completely monotone on (0, 4], and
x
-
y
is the Euclidean distance between points
x
and
y
. We show that any sharp point configuration
ω
¯
N
on
S
d
, which is antipodal or is a spherical design of an even strength is a solution to this problem. We also prove that the absolute maximum over
S
d
of the potential of any such configuration
ω
¯
N
is attained at points of
ω
¯
N
.</description><subject>Analysis</subject><subject>Configuration management</subject><subject>Euclidean geometry</subject><subject>Mathematics</subject><subject>Mathematics and Statistics</subject><subject>Numerical Analysis</subject><issn>0176-4276</issn><issn>1432-0940</issn><fulltext>true</fulltext><rsrctype>article</rsrctype><creationdate>2024</creationdate><recordtype>article</recordtype><recordid>eNp9kE1LxDAQhoMouK7-AU8Bz9GZJE3boxS_YFeF1XNIt8lulzWpSRfUX2_dCt48DTM87zvwEHKOcIkA-VUCECpjwAWDUilkeEAmKAUfVgmHZAKYKyZ5ro7JSUobAMwKkU_I47z1bG4-6HPYmth-mb4NnroQaWVjb1pPq61JySYaHF2sTexoFbxrV7u4R4e7p_3a0kW3ttGekiNntsme_c4peb29eanu2ezp7qG6nrGlwLJnLi9qQNVwiYKDBaOWNudFySVvbFbXSmIJvC6cNVzJWtbQcCGbzJVDTnAUU3Ix9nYxvO9s6vUm7KIfXmqBWKqMlwoGio_UMoaUonW6i-2biZ8aQf9406M3PXjTe2_6p1qMoTTAfmXjX_U_qW_aS27p</recordid><startdate>20241001</startdate><enddate>20241001</enddate><creator>Borodachov, Sergiy</creator><general>Springer US</general><general>Springer Nature B.V</general><scope>AAYXX</scope><scope>CITATION</scope></search><sort><creationdate>20241001</creationdate><title>Min-Max Polarization for Certain Classes of Sharp Configurations on the Sphere</title><author>Borodachov, Sergiy</author></sort><facets><frbrtype>5</frbrtype><frbrgroupid>cdi_FETCH-LOGICAL-c319t-f78b016d241320e0a6ce7289242de5bb641902b8fea264b4b0d234d5f98b03213</frbrgroupid><rsrctype>articles</rsrctype><prefilter>articles</prefilter><language>eng</language><creationdate>2024</creationdate><topic>Analysis</topic><topic>Configuration management</topic><topic>Euclidean geometry</topic><topic>Mathematics</topic><topic>Mathematics and Statistics</topic><topic>Numerical Analysis</topic><toplevel>peer_reviewed</toplevel><toplevel>online_resources</toplevel><creatorcontrib>Borodachov, Sergiy</creatorcontrib><collection>CrossRef</collection><jtitle>Constructive approximation</jtitle></facets><delivery><delcategory>Remote Search Resource</delcategory><fulltext>fulltext</fulltext></delivery><addata><au>Borodachov, Sergiy</au><format>journal</format><genre>article</genre><ristype>JOUR</ristype><atitle>Min-Max Polarization for Certain Classes of Sharp Configurations on the Sphere</atitle><jtitle>Constructive approximation</jtitle><stitle>Constr Approx</stitle><date>2024-10-01</date><risdate>2024</risdate><volume>60</volume><issue>2</issue><spage>237</spage><epage>252</epage><pages>237-252</pages><issn>0176-4276</issn><eissn>1432-0940</eissn><abstract>We consider the problem of finding an
N
-point configuration on the sphere
S
d
⊂
R
d
+
1
with the smallest absolute maximum value over
S
d
of its total potential. The potential induced by each point
y
in a given configuration at a point
x
∈
S
d
is
f
x
-
y
2
, where
f
is continuous on [0, 4] and completely monotone on (0, 4], and
x
-
y
is the Euclidean distance between points
x
and
y
. We show that any sharp point configuration
ω
¯
N
on
S
d
, which is antipodal or is a spherical design of an even strength is a solution to this problem. We also prove that the absolute maximum over
S
d
of the potential of any such configuration
ω
¯
N
is attained at points of
ω
¯
N
.</abstract><cop>New York</cop><pub>Springer US</pub><doi>10.1007/s00365-023-09661-1</doi><tpages>16</tpages></addata></record> |
| fulltext | fulltext |
| identifier | ISSN: 0176-4276 |
| ispartof | Constructive approximation, 2024-10, Vol.60 (2), p.237-252 |
| issn | 0176-4276 1432-0940 |
| language | eng |
| recordid | cdi_proquest_journals_3119652960 |
| source | Springer Nature - Complete Springer Journals |
| subjects | Analysis Configuration management Euclidean geometry Mathematics Mathematics and Statistics Numerical Analysis |
| title | Min-Max Polarization for Certain Classes of Sharp Configurations on the Sphere |
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