Study on the convexity of intersection bodies of a convex body with radial centers
| Project/Area Number |
17K14191
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| Research Category |
Grant-in-Aid for Young Scientists (B)
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| Allocation Type | Multi-year Fund |
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| Research Field |
Geometry
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| Research Institution | Fukuoka University (2019-2022)
University of Miyazaki (2017-2018) |
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Principal Investigator |
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| Project Period (FY) |
2017-04-01 – 2023-03-31
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| Project Status |
Completed (Fiscal Year 2022)
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| 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)
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| 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 |
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| 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.
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| 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。
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Report
(7 results)
Research Products
(20 results)
