• Search Research Projects
  • Search Researchers
  • How to Use
  1. Back to previous page

Study on the convexity of intersection bodies of a convex body with radial centers

Research Project

Project/Area Number 17K14191
Research Category

Grant-in-Aid for Young Scientists (B)

Allocation TypeMulti-year Fund
Research Field Geometry
Research InstitutionFukuoka University (2019-2022)
University of Miyazaki (2017-2018)

Principal Investigator

Sakata Shigehiro  福岡大学, 理学部, 准教授 (30732937)

Project Period (FY) 2017-04-01 – 2023-03-31
Project Status Completed (Fiscal Year 2022)
Budget Amount *help
¥4,290,000 (Direct Cost: ¥3,300,000、Indirect Cost: ¥990,000)
Fiscal Year 2019: ¥1,170,000 (Direct Cost: ¥900,000、Indirect Cost: ¥270,000)
Fiscal Year 2018: ¥1,430,000 (Direct Cost: ¥1,100,000、Indirect Cost: ¥330,000)
Fiscal Year 2017: ¥1,690,000 (Direct Cost: ¥1,300,000、Indirect Cost: ¥390,000)
Keywords凸体 / 交差体 / 凸性 / Busemannの定理 / Busemann-Pettyの問題 / 輻射中心 / 三角形 / 動径関数 / モーメント / 凸多角形 / たたみ込み / 正三角形 / 平行体
Outline of Final Research Achievements

The setting of this research is in the Euclidean space, and the object is an intersection body. An intersection body is a star body made from a star body. In general, the intersection bodies of a convex body containing the origin is not convex. Busemann’s theorem states that the intersection body of any centered convex body is convex. We are interested in how to construct convex intersection bodies from convex bodies without any symmetry (especially, central symmetry).
We showed the following. Let L be a star body such that its radial function is twice continuously differentiable. Let K be the radial sum of L and a centered ball. If the radius of the ball is “large enough”, then K and the intersection body of K are convex.

Academic Significance and Societal Importance of the Research Achievements

本研究はBusemann-Pettyの問題に由来する。Busemann-Pettyの問題の心は「2つの凸体の体積は、平面による切り口の面積によって比べられるか」であり、凸幾何学の逆問題(geometric tomography)における中心的課題の1つとなっている。Busemann-Pettyの問題の解は凸な交差体であることが知られている。そのため、凸な交差体の構成は重要な課題である。
本研究では、凸な交差体の新しい構成方法を提示した。本研究成果の応用例として、Busemann-Pettyの問題の解の具体的な構成が期待される。

Report

(7 results)
  • 2022 Annual Research Report   Final Research Report ( PDF )
  • 2021 Research-status Report
  • 2020 Research-status Report
  • 2019 Research-status Report
  • 2018 Research-status Report
  • 2017 Research-status Report
  • Research Products

    (20 results)

All 2022 2021 2020 2019 2018 2017 Other

All Int'l Joint Research (5 results) Journal Article (5 results) (of which Int'l Joint Research: 1 results,  Peer Reviewed: 5 results) Presentation (9 results) (of which Int'l Joint Research: 2 results,  Invited: 8 results) Remarks (1 results)

  • [Int'l Joint Research] ワルシャワ工科大学(ポーランド)

    • Related Report
      2021 Research-status Report
  • [Int'l Joint Research] ワルシャワ工科大学(ポーランド)

    • Related Report
      2020 Research-status Report
  • [Int'l Joint Research] ワルシャワ工科大学(ポーランド)

    • Related Report
      2019 Research-status Report
  • [Int'l Joint Research] ワルシャワ工科大学/ワルシャワ大学/ヤギエウォ大学(ポーランド)

    • Related Report
      2018 Research-status Report
  • [Int'l Joint Research] ワルシャワ工科大学/ワルシャワ大学(ポーランド)

    • Related Report
      2017 Research-status Report
  • [Journal Article] Euclidean geometric description of radial centers of a triangle2022

    • Author(s)
      Sakata Shigehiro
    • Journal Title

      Colloquium Mathematicum

      Volume: 170 Issue: 2 Pages: 275-288

    • DOI

      10.4064/cm8493-12-2021

    • Related Report
      2022 Annual Research Report
    • Peer Reviewed
  • [Journal Article] An extremum problem for the power moment of a convex polygon contained in a disc2021

    • Author(s)
      Herburt Irmina、Sakata Shigehiro
    • Journal Title

      Advances in Geometry

      Volume: 21 Issue: 4 Pages: 599-609

    • DOI

      10.1515/advgeom-2021-0021

    • Related Report
      2021 Research-status Report
    • Peer Reviewed / Int'l Joint Research
  • [Journal Article] Analytic characterization of equilateral triangles2021

    • Author(s)
      Shigehiro Sakata
    • Journal Title

      Annali di Matematica Pura ed Applicata

      Volume: - Issue: 5 Pages: 2191-2212

    • DOI

      10.1007/s10231-021-01075-9

    • Related Report
      2020 Research-status Report
    • Peer Reviewed
  • [Journal Article] Movement of time-delayed hot spots in Euclidean space for a degenerate initial state2020

    • Author(s)
      Sakata Shigehiro、Wakasugi Yuta
    • Journal Title

      Discrete & Continuous Dynamical Systems - A

      Volume: 40 Issue: 5 Pages: 2705-2738

    • DOI

      10.3934/dcds.2020147

    • Related Report
      2019 Research-status Report
    • Peer Reviewed
  • [Journal Article] Stationary radial centers and symmetry of convex polytopes2019

    • Author(s)
      Shigehiro Sakata
    • Journal Title

      Colloquium Mathematicum

      Volume: 159 Issue: 1 Pages: 91-106

    • DOI

      10.4064/cm7712-11-2018

    • Related Report
      2019 Research-status Report 2018 Research-status Report
    • Peer Reviewed
  • [Presentation] 距離核ポテンシャルの臨界点と正三角形の特徴づけ2020

    • Author(s)
      坂田繁洋
    • Organizer
      広島数理解析セミナー・冬の研究会2020
    • Related Report
      2019 Research-status Report
    • Invited
  • [Presentation] 凸多角形のモーメントの極値問題2019

    • Author(s)
      坂田繁洋
    • Organizer
      福岡大学微分幾何セミナー
    • Related Report
      2019 Research-status Report
    • Invited
  • [Presentation] 距離核ポテンシャルの臨界点による正三角形の特徴づけ2019

    • Author(s)
      坂田繁洋
    • Organizer
      第66回幾何学シンポジウム
    • Related Report
      2019 Research-status Report
    • Invited
  • [Presentation] 凸体のRieszポテンシャル, 中心, 切り口と影2019

    • Author(s)
      坂田繁洋
    • Organizer
      福岡大学微分幾何研究集会2019
    • Related Report
      2019 Research-status Report
    • Invited
  • [Presentation] Critical points of Riesz potentials and characterization of regular triangles2018

    • Author(s)
      Shigehiro Sakata
    • Organizer
      Seminarium "Geometria Przestrzeni Banach"
    • Related Report
      2018 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] Symmetry of a triangle and critical points of Riesz potential2018

    • Author(s)
      坂田繁洋
    • Organizer
      首都大学東京幾何セミナー
    • Related Report
      2017 Research-status Report
    • Invited
  • [Presentation] Characterization of regular triangles in terms of critical points of Riesz potentials2017

    • Author(s)
      Shigehiro Sakata
    • Organizer
      Convex, Discrete and Integral Geometry
    • Related Report
      2017 Research-status Report
    • Int'l Joint Research
  • [Presentation] ポテンシャルの臨界点と凸多角形の対称性2017

    • Author(s)
      坂田繁洋
    • Organizer
      幾何セミナー(熊本)
    • Related Report
      2017 Research-status Report
    • Invited
  • [Presentation] 関数の最大点と凸体の対称性2017

    • Author(s)
      坂田繁洋
    • Organizer
      第64回幾何学シンポジウム
    • Related Report
      2017 Research-status Report
    • Invited
  • [Remarks] Shigehiro Sakata's page

    • URL

      https://sites.google.com/site/shigehirosakata/

    • Related Report
      2022 Annual Research Report 2021 Research-status Report 2020 Research-status Report 2019 Research-status Report 2018 Research-status Report 2017 Research-status Report

URL: 

Published: 2017-04-28   Modified: 2024-01-30  

Information User Guide FAQ News Terms of Use Attribution of KAKENHI

Powered by NII kakenhi