Dirac masses and isometric rigidity

Geher GP, Titkos T, Virosztek D. 2019. Dirac masses and isometric rigidity. Kyoto RIMS Kôkyûroku. Research on isometries as preserver problems and related topics vol. 2125, 34–41.

Conference Paper | Published | English
Author
Geher, Gyorgy Pal; Titkos, Tamas; Virosztek, DanielISTA

Corresponding author has ISTA affiliation

Department
Abstract
The aim of this short note is to expound one particular issue that was discussed during the talk [10] given at the symposium ”Researches on isometries as preserver problems and related topics” at Kyoto RIMS. That is, the role of Dirac masses by describing the isometry group of various metric spaces of probability measures. This article is of survey character, and it does not contain any essentially new results.From an isometric point of view, in some cases, metric spaces of measures are similar to C(K)-type function spaces. Similarity means here that their isometries are driven by some nice transformations of the underlying space. Of course, it depends on the particular choice of the metric how nice these transformations should be. Sometimes, as we will see, being a homeomorphism is enough to generate an isometry. But sometimes we need more: the transformation must preserve the underlying distance as well. Statements claiming that isometries in questions are necessarily induced by homeomorphisms are called Banach-Stone-type results, while results asserting that the underlying transformation is necessarily an isometry are termed as isometric rigidity results.As Dirac masses can be considered as building bricks of the set of all Borel measures, a natural question arises:Is it enough to understand how an isometry acts on the set of Dirac masses? Does this action extend uniquely to all measures?In what follows, we will thoroughly investigate this question.
Publishing Year
Date Published
2019-01-30
Proceedings Title
Kyoto RIMS Kôkyûroku
Publisher
Research Institute for Mathematical Sciences, Kyoto University
Acknowledgement
This paper is part of a long term collaboration investigating the isometric structure of Wasserstein spaces. The authors would like to thank the warm hospitality and generosity of László Erdós and his group at Institute of Science and Technology Austria. T. Titkos wants to thank Oriental Business and Innovation Center ‐ OBIC for providing financial support to participate in the symposium at the Kyoto RIMS. Gy. P. Gehér was supported by the Leverhulme Trust Early Career Fellowship (ECF‐2018‐125), and also by the Hungarian National Research, Development and Innovation Office (K115383). T. Titkos was supported by the Hungarian National Research, Development and Innovation Office‐ NKFIH (PD128374), by the János Bolyai Research Scholarship of the Hungarian Academy of Sciences, and by the UNKP‐18‐4‐BGE‐3 New National Excellence Program of the Ministry of Human Capacities. D. Virosztek was supported by the ISTFELLOW program of the Institute of Science and Technology Austria (project code IC1027FELL01 ) and partially supported by the Hungarian National Research, Development and Innovation Office NKFIH (grant no. K124152 and grant no. KH129601)
Volume
2125
Page
34-41
Conference
Research on isometries as preserver problems and related topics
Conference Location
Kyoto, Japan
Conference Date
2019-01-28 – 2019-01-30
IST-REx-ID

Cite this

Geher GP, Titkos T, Virosztek D. Dirac masses and isometric rigidity. In: Kyoto RIMS Kôkyûroku. Vol 2125. Research Institute for Mathematical Sciences, Kyoto University; 2019:34-41.
Geher, G. P., Titkos, T., & Virosztek, D. (2019). Dirac masses and isometric rigidity. In Kyoto RIMS Kôkyûroku (Vol. 2125, pp. 34–41). Kyoto, Japan: Research Institute for Mathematical Sciences, Kyoto University.
Geher, Gyorgy Pal, Tamas Titkos, and Daniel Virosztek. “Dirac Masses and Isometric Rigidity.” In Kyoto RIMS Kôkyûroku, 2125:34–41. Research Institute for Mathematical Sciences, Kyoto University, 2019.
G. P. Geher, T. Titkos, and D. Virosztek, “Dirac masses and isometric rigidity,” in Kyoto RIMS Kôkyûroku, Kyoto, Japan, 2019, vol. 2125, pp. 34–41.
Geher GP, Titkos T, Virosztek D. 2019. Dirac masses and isometric rigidity. Kyoto RIMS Kôkyûroku. Research on isometries as preserver problems and related topics vol. 2125, 34–41.
Geher, Gyorgy Pal, et al. “Dirac Masses and Isometric Rigidity.” Kyoto RIMS Kôkyûroku, vol. 2125, Research Institute for Mathematical Sciences, Kyoto University, 2019, pp. 34–41.
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