Geometric study of potentials and optimal art gallery problem

2016 Fiscal Year Final Research Report

• Project/Area Number: 25610014
• Research Category: Grant-in-Aid for Challenging Exploratory Research
• Allocation Type: Multi-year Fund
• Research Field: Geometry
• Research Institution: Chiba University (2015-2016); Tokyo Metropolitan University (2013-2014)
• Principal Investigator: Imai Jun 千葉大学, 大学院理学研究科, 教授 (70221132)
• Co-Investigator(Renkei-kenkyūsha): HAMADA TATSUYOSHI 日本大学, 生物資源学部, 准教授 (90299537)
• Project Period (FY): 2013-04-01 – 2017-03-31
• Keywords: Riesz ポテンシャル / 美術館問題 / プログラミング

Outline of Final Research Achievements

(1) Any critical point of a potential with kernel being a monotone function of the distance is included in a minimal unfolded region. Some geometric properties of the minimal unfolded regions have been given. (2) Regularization of the Riesz potential and Riesz energy of a submanifold of the Euclidean space is given. Some of the residues of the Riesz energy thus obtained of a compact body have been computed (joint work with Gil Solanes). (3) A program for the optimal art-gallery problem, which seeks for an optimal position of cameras to monitor a given gallery, has been obtained via outsourcing. (4) A program that can deform a given knot in the unit 3-sphere to decrease the energy has been obtained by outsourcing to Wolfram.

Free Research Field

幾何学