Brunn-Minkowski inequality for $\theta$-convolution bodies via Ball's bodies
| Title: | Brunn-Minkowski inequality for $\theta$-convolution bodies via Ball's bodies |
|---|---|
| Authors: | Alonso-Gutiérrez, David, Goñi, Javier Martín |
| Publication Year: | 2022 |
| Collection: | Mathematics |
| Subject Terms: | Mathematics - Metric Geometry, 52A40 |
| More Details: | We consider the problem of finding the best function $\varphi_n:[0,1]\to\mathbb{R}$ such that for any pair of convex bodies $K,L\in\mathbb{R}^n$ the following Brunn-Minkowski type inequality holds $$ |K+_\theta L|^\frac{1}{n}\geq\varphi_n(\theta)(|K|^\frac{1}{n}+|L|^\frac{1}{n}), $$ where $K+_\theta L$ is the $\theta$-convolution body of $K$ and $L$. We prove a sharp inclusion of the family of Ball's bodies of an $\alpha$-concave function in its super-level sets in order to provide the best possible function in the range $\left(\frac{3}{4}\right)^n\leq\theta\leq1$, characterizing the equality cases. |
| Document Type: | Working Paper |
| Access URL: | http://arxiv.org/abs/2211.17069 |
| Accession Number: | edsarx.2211.17069 |
| Database: | arXiv |
| FullText | Text: Availability: 0 CustomLinks: – Url: http://arxiv.org/abs/2211.17069 Name: EDS - Arxiv Category: fullText Text: View this record from Arxiv MouseOverText: View this record from Arxiv – Url: https://resolver.ebsco.com/c/xy5jbn/result?sid=EBSCO:edsarx&genre=article&issn=&ISBN=&volume=&issue=&date=20221130&spage=&pages=&title=Brunn-Minkowski inequality for $\theta$-convolution bodies via Ball's bodies&atitle=Brunn-Minkowski%20inequality%20for%20%24%5Ctheta%24-convolution%20bodies%20via%20Ball%27s%20bodies&aulast=Alonso-Guti%C3%A9rrez%2C%20David&id=DOI: Name: Full Text Finder (for New FTF UI) (s8985755) Category: fullText Text: Find It @ SCU Libraries MouseOverText: Find It @ SCU Libraries |
|---|---|
| Header | DbId: edsarx DbLabel: arXiv An: edsarx.2211.17069 RelevancyScore: 1037 AccessLevel: 3 PubType: Report PubTypeId: report PreciseRelevancyScore: 1037.18481445313 |
| IllustrationInfo | |
| Items | – Name: Title Label: Title Group: Ti Data: Brunn-Minkowski inequality for $\theta$-convolution bodies via Ball's bodies – Name: Author Label: Authors Group: Au Data: <searchLink fieldCode="AR" term="%22Alonso-Gutiérrez%2C+David%22">Alonso-Gutiérrez, David</searchLink><br /><searchLink fieldCode="AR" term="%22Goñi%2C+Javier+Martín%22">Goñi, Javier Martín</searchLink> – Name: DatePubCY Label: Publication Year Group: Date Data: 2022 – Name: Subset Label: Collection Group: HoldingsInfo Data: Mathematics – Name: Subject Label: Subject Terms Group: Su Data: <searchLink fieldCode="DE" term="%22Mathematics+-+Metric+Geometry%22">Mathematics - Metric Geometry</searchLink><br /><searchLink fieldCode="DE" term="%2252A40%22">52A40</searchLink> – Name: Abstract Label: Description Group: Ab Data: We consider the problem of finding the best function $\varphi_n:[0,1]\to\mathbb{R}$ such that for any pair of convex bodies $K,L\in\mathbb{R}^n$ the following Brunn-Minkowski type inequality holds $$ |K+_\theta L|^\frac{1}{n}\geq\varphi_n(\theta)(|K|^\frac{1}{n}+|L|^\frac{1}{n}), $$ where $K+_\theta L$ is the $\theta$-convolution body of $K$ and $L$. We prove a sharp inclusion of the family of Ball's bodies of an $\alpha$-concave function in its super-level sets in order to provide the best possible function in the range $\left(\frac{3}{4}\right)^n\leq\theta\leq1$, characterizing the equality cases. – Name: TypeDocument Label: Document Type Group: TypDoc Data: Working Paper – Name: URL Label: Access URL Group: URL Data: <link linkTarget="URL" linkTerm="http://arxiv.org/abs/2211.17069" linkWindow="_blank">http://arxiv.org/abs/2211.17069</link> – Name: AN Label: Accession Number Group: ID Data: edsarx.2211.17069 |
| PLink | https://login.libproxy.scu.edu/login?url=https://search.ebscohost.com/login.aspx?direct=true&site=eds-live&scope=site&db=edsarx&AN=edsarx.2211.17069 |
| RecordInfo | BibRecord: BibEntity: Subjects: – SubjectFull: Mathematics - Metric Geometry Type: general – SubjectFull: 52A40 Type: general Titles: – TitleFull: Brunn-Minkowski inequality for $\theta$-convolution bodies via Ball's bodies Type: main BibRelationships: HasContributorRelationships: – PersonEntity: Name: NameFull: Alonso-Gutiérrez, David – PersonEntity: Name: NameFull: Goñi, Javier Martín IsPartOfRelationships: – BibEntity: Dates: – D: 30 M: 11 Type: published Y: 2022 |
| ResultId | 1 |