On Grünbaum Type Inequalities
Abstract Given a compact set $$K\subset {\mathbb {R}}^n$$ of positive volume, and fixing a hyperplane H passing through its centroid, we find a sharp lower bound for the ratio $$\mathrm {vol}(K^{-})/\mathrm {vol}(K)$$, depending on the concavity nature of the function that gives the volumes of cross...
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| Autor*in: |
Marín Sola, Francisco [verfasserIn] |
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| Format: |
Artikel |
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| Sprache: |
Englisch |
| Erschienen: |
2021 |
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| Anmerkung: |
© Mathematica Josephina, Inc. 2021 |
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| Übergeordnetes Werk: |
Enthalten in: The journal of geometric analysis - Springer US, 1991, 31(2021), 10 vom: 16. März, Seite 9981-9995 |
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| Übergeordnetes Werk: |
volume:31 ; year:2021 ; number:10 ; day:16 ; month:03 ; pages:9981-9995 |
| Links: |
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| DOI / URN: |
10.1007/s12220-021-00635-y |
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OLC2127394151 |
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| 520 | |a Abstract Given a compact set $$K\subset {\mathbb {R}}^n$$ of positive volume, and fixing a hyperplane H passing through its centroid, we find a sharp lower bound for the ratio $$\mathrm {vol}(K^{-})/\mathrm {vol}(K)$$, depending on the concavity nature of the function that gives the volumes of cross-sections (parallel to H) of K, where $$K^{-}$$ denotes the intersection of K with a halfspace bounded by H. When K is convex, this inequality recovers a classical result by Grünbaum. To this respect, we also show that the log-concave case is the limit concavity assumption for such a generalization of Grünbaum’s inequality. | ||
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10.1007/s12220-021-00635-y doi (DE-627)OLC2127394151 (DE-He213)s12220-021-00635-y-p DE-627 ger DE-627 rakwb eng 510 VZ 17,1 ssgn Marín Sola, Francisco verfasserin aut On Grünbaum Type Inequalities 2021 Text txt rdacontent ohne Hilfsmittel zu benutzen n rdamedia Band nc rdacarrier © Mathematica Josephina, Inc. 2021 Abstract Given a compact set $$K\subset {\mathbb {R}}^n$$ of positive volume, and fixing a hyperplane H passing through its centroid, we find a sharp lower bound for the ratio $$\mathrm {vol}(K^{-})/\mathrm {vol}(K)$$, depending on the concavity nature of the function that gives the volumes of cross-sections (parallel to H) of K, where $$K^{-}$$ denotes the intersection of K with a halfspace bounded by H. When K is convex, this inequality recovers a classical result by Grünbaum. To this respect, we also show that the log-concave case is the limit concavity assumption for such a generalization of Grünbaum’s inequality. Grünbaum’s inequality -Concave function Log-concave function Brunn’s concavity principle Sections of convex bodies Centroid Yepes Nicolás, Jesús (orcid)0000-0001-9815-1736 aut Enthalten in The journal of geometric analysis Springer US, 1991 31(2021), 10 vom: 16. März, Seite 9981-9995 (DE-627)131006398 (DE-600)1086949-9 (DE-576)028039211 1050-6926 nnns volume:31 year:2021 number:10 day:16 month:03 pages:9981-9995 https://doi.org/10.1007/s12220-021-00635-y lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_OLC SSG-OLC-MAT SSG-OPC-MAT GBV_ILN_2409 AR 31 2021 10 16 03 9981-9995 |
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10.1007/s12220-021-00635-y doi (DE-627)OLC2127394151 (DE-He213)s12220-021-00635-y-p DE-627 ger DE-627 rakwb eng 510 VZ 17,1 ssgn Marín Sola, Francisco verfasserin aut On Grünbaum Type Inequalities 2021 Text txt rdacontent ohne Hilfsmittel zu benutzen n rdamedia Band nc rdacarrier © Mathematica Josephina, Inc. 2021 Abstract Given a compact set $$K\subset {\mathbb {R}}^n$$ of positive volume, and fixing a hyperplane H passing through its centroid, we find a sharp lower bound for the ratio $$\mathrm {vol}(K^{-})/\mathrm {vol}(K)$$, depending on the concavity nature of the function that gives the volumes of cross-sections (parallel to H) of K, where $$K^{-}$$ denotes the intersection of K with a halfspace bounded by H. When K is convex, this inequality recovers a classical result by Grünbaum. To this respect, we also show that the log-concave case is the limit concavity assumption for such a generalization of Grünbaum’s inequality. Grünbaum’s inequality -Concave function Log-concave function Brunn’s concavity principle Sections of convex bodies Centroid Yepes Nicolás, Jesús (orcid)0000-0001-9815-1736 aut Enthalten in The journal of geometric analysis Springer US, 1991 31(2021), 10 vom: 16. März, Seite 9981-9995 (DE-627)131006398 (DE-600)1086949-9 (DE-576)028039211 1050-6926 nnns volume:31 year:2021 number:10 day:16 month:03 pages:9981-9995 https://doi.org/10.1007/s12220-021-00635-y lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_OLC SSG-OLC-MAT SSG-OPC-MAT GBV_ILN_2409 AR 31 2021 10 16 03 9981-9995 |
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10.1007/s12220-021-00635-y doi (DE-627)OLC2127394151 (DE-He213)s12220-021-00635-y-p DE-627 ger DE-627 rakwb eng 510 VZ 17,1 ssgn Marín Sola, Francisco verfasserin aut On Grünbaum Type Inequalities 2021 Text txt rdacontent ohne Hilfsmittel zu benutzen n rdamedia Band nc rdacarrier © Mathematica Josephina, Inc. 2021 Abstract Given a compact set $$K\subset {\mathbb {R}}^n$$ of positive volume, and fixing a hyperplane H passing through its centroid, we find a sharp lower bound for the ratio $$\mathrm {vol}(K^{-})/\mathrm {vol}(K)$$, depending on the concavity nature of the function that gives the volumes of cross-sections (parallel to H) of K, where $$K^{-}$$ denotes the intersection of K with a halfspace bounded by H. When K is convex, this inequality recovers a classical result by Grünbaum. To this respect, we also show that the log-concave case is the limit concavity assumption for such a generalization of Grünbaum’s inequality. Grünbaum’s inequality -Concave function Log-concave function Brunn’s concavity principle Sections of convex bodies Centroid Yepes Nicolás, Jesús (orcid)0000-0001-9815-1736 aut Enthalten in The journal of geometric analysis Springer US, 1991 31(2021), 10 vom: 16. März, Seite 9981-9995 (DE-627)131006398 (DE-600)1086949-9 (DE-576)028039211 1050-6926 nnns volume:31 year:2021 number:10 day:16 month:03 pages:9981-9995 https://doi.org/10.1007/s12220-021-00635-y lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_OLC SSG-OLC-MAT SSG-OPC-MAT GBV_ILN_2409 AR 31 2021 10 16 03 9981-9995 |
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Abstract Given a compact set $$K\subset {\mathbb {R}}^n$$ of positive volume, and fixing a hyperplane H passing through its centroid, we find a sharp lower bound for the ratio $$\mathrm {vol}(K^{-})/\mathrm {vol}(K)$$, depending on the concavity nature of the function that gives the volumes of cross-sections (parallel to H) of K, where $$K^{-}$$ denotes the intersection of K with a halfspace bounded by H. When K is convex, this inequality recovers a classical result by Grünbaum. To this respect, we also show that the log-concave case is the limit concavity assumption for such a generalization of Grünbaum’s inequality. © Mathematica Josephina, Inc. 2021 |
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Abstract Given a compact set $$K\subset {\mathbb {R}}^n$$ of positive volume, and fixing a hyperplane H passing through its centroid, we find a sharp lower bound for the ratio $$\mathrm {vol}(K^{-})/\mathrm {vol}(K)$$, depending on the concavity nature of the function that gives the volumes of cross-sections (parallel to H) of K, where $$K^{-}$$ denotes the intersection of K with a halfspace bounded by H. When K is convex, this inequality recovers a classical result by Grünbaum. To this respect, we also show that the log-concave case is the limit concavity assumption for such a generalization of Grünbaum’s inequality. © Mathematica Josephina, Inc. 2021 |
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Abstract Given a compact set $$K\subset {\mathbb {R}}^n$$ of positive volume, and fixing a hyperplane H passing through its centroid, we find a sharp lower bound for the ratio $$\mathrm {vol}(K^{-})/\mathrm {vol}(K)$$, depending on the concavity nature of the function that gives the volumes of cross-sections (parallel to H) of K, where $$K^{-}$$ denotes the intersection of K with a halfspace bounded by H. When K is convex, this inequality recovers a classical result by Grünbaum. To this respect, we also show that the log-concave case is the limit concavity assumption for such a generalization of Grünbaum’s inequality. © Mathematica Josephina, Inc. 2021 |
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2024-07-04T10:39:46.145Z |
| _version_ |
1803644657884725248 |
| fullrecord_marcxml |
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