Relation between measure concentration inequality and isoperimetric inequality under the curvature dimension condition

2014 Fiscal Year Final Research Report

• Project/Area Number: 24740042
• Research Category: Grant-in-Aid for Young Scientists (B)
• Allocation Type: Multi-year Fund
• Research Field: Geometry
• Research Institution: Nagoya University
• Principal Investigator: ASUKA TAKATSU 名古屋大学, 多元数理科学研究科, 助教 (90623554)
• Project Period (FY): 2012-04-01 – 2015-03-31
• Keywords: 測度距離空間 / 最適輸送理論 / 情報幾何 / 測度の集中減少 / 等周不等式

Outline of Final Research Achievements

I investigated \phi-Gaussian measures, which is obtained by generalizing the Gaussian measure via a positive continuous non-decreasing function \phi on the positive half line. If \phi is the identity function then the \phi-Gaussian measure recovers the Gaussian measure,and in the case that \phi is the power function with exponent q which lies in (0,1) or (1,2), the \phi-Gaussian measure is a power distribution, so-called the q-Gaussian measure.
In the joint work with Shin-ichi Ohta, we provided a useful condition to we analyze a metric measure space based on the theory of the q- Gaussian measure. By using this condition, we obtained analytic/geometric results such as a set of functional inequalities (e.g. a transport inequality), the concentration of measure phenomenon (which describes a relation between volume and distance) and the gradient flow of the \phi-divergence on the Wasserstein space.

Free Research Field

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