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DISTANCE FUNCTIONS ON CONVEX BODIES AND SYMPLECTIC TORIC MANIFOLDS

Part of: Symplectic geometry, contact geometry Global differential geometry Polytopes and polyhedra

Published online by Cambridge University Press:  02 April 2025

HAJIME FUJITA Open the ORCID record for HAJIME FUJITA [Opens in a new window] ,
YU KITABEPPU Open the ORCID record for YU KITABEPPU [Opens in a new window]  and
AYATO MITSUISHI Open the ORCID record for AYATO MITSUISHI [Opens in a new window]
HAJIME FUJITA*
Affiliation:
Faculty of Science Japan Women’s University Bunkyo City Tokyo Japan
YU KITABEPPU
Affiliation:
Faculty of Advanced Science and Technology Kumamoto University Chuo Ward Kumamoto Japan [email protected]
AYATO MITSUISHI
Affiliation:
Faculty of Science Fukuoka University Jonan Ward Fukuoka Japan [email protected]
*
[email protected]
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Abstract

In this paper we discuss three distance functions on the set of convex bodies. In particular we study the convergence of Delzant polytopes, which are fundamental objects in symplectic toric geometry. By using these observations, we derive some convergence theorems for symplectic toric manifolds with respect to the Gromov–Hausdorff distance.

Keywords

Delzant polytopesymplectic toric manifoldGromov–Hausdorff convergence

MSC classification

Primary: 53C23: Global geometric and topological methods (à la Gromov); differential geometric analysis on metric spaces
Secondary: 53D20: Momentum maps; symplectic reduction 52B12: Special polytopes (linear programming, centrally symmetric, etc.)
Type
Article
Information
Nagoya Mathematical Journal , First View , pp. 1 - 23
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Copyright
© The Author(s), 2025. Published by Cambridge University Press on behalf of Foundation Nagoya Mathematical Journal

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References

Abreu, M., “Kähler geometry of toric manifolds in symplectic coordinates” in Y. Eliashberg, B. A. Khesin and F. Lalonde (eds.), Symplectic and Contact Topology: Interactions and Perspectives (Toronto, ON/Montreal, QC, 2001), volume 35 of Fields Institute Communications, American Mathematical Society, Providence, RI, 2003, pp. 1–24.Google Scholar
Ambrosio, L., Gigli, N. and Savaré, G., Gradient flows in metric spaces and in the space of probability measures, second edition, Lectures in Mathematics ETH Zürich, Birkhäuser Verlag, Basel, 2008.Google Scholar
Buchstaber, V. M. and Panov, T. E., Toric topology, volume 204 of Mathematical Surveys and Monographs, American Mathematical Society, Providence, RI, 2015.CrossRefGoogle Scholar
Delzant, T., Hamiltoniens périodiques et images convexes de l’application moment , Bull. Soc. Math. France 116 (1988), no. 3, 315–339.CrossRefGoogle Scholar
Dinghas, A., Über das Verhalten der Entfernung zweier Punktmengen bei gleichzeitiger Symmetrisierung derselben , Arch. Math. (Basel) 8 (1957), 46–51.CrossRefGoogle Scholar
Fujita, H. and Ohashi, K., A metric on the moduli space of bodies, preprint, arXiv:1804.05161, 2018.Google Scholar
Fukaya, K., Theory of convergence for Riemannian orbifolds , Japan. J. Math. (N.S.) 12 (1986), no. 1, 121–160.CrossRefGoogle Scholar
Garling, D. J. H., Analysis on polish spaces and an introduction to optimal transportation, volume 89 of London Mathematical Society Student Texts, Cambridge University Press, Cambridge, 2018.Google Scholar
Guillemin, V., Kaehler structures on toric varieties , J. Differ. Geom. 40 (1994), no. 2, 285–309.CrossRefGoogle Scholar
Harada, M. and Kaveh, K., Integrable systems, toric degenerations and Okounkov bodies , Invent. Math. 202 (2015), no. 3, 927–985.CrossRefGoogle Scholar
Iriyeh, H. and Ono, H., Almost all Lagrangian torus orbits in $\mathbb{C}{P}^n$ are not Hamiltonian volume minimizing , Ann. Global Anal. Geom. 50 (2016), no. 1, 85–96.CrossRefGoogle Scholar
Karshon, Y., Kessler, L. and Pinsonnault, M., A compact symplectic four-manifold admits only finitely many inequivalent toric actions , J. Symp. Geom. 5 (2007), no. 2, 139–166.CrossRefGoogle Scholar
Nishinou, T., Nohara, Y. and Ueda, K. Toric degenerations of Gel’fand-Cetlin systems and potential functions , Adv. Math. 224 (2010), no. 2, 648–706.CrossRefGoogle Scholar
Pelayo, A., Pires, A. R., Ratiu, T. S. and Sabatini, S., Moduli spaces of toric manifolds , Geom. Dedic. 169 (2014), 323–341.CrossRefGoogle Scholar
Shephard, G. C. and Webster, R. J., Metrics for sets of convex bodies , Mathematika 12 (1965), 73–88.CrossRefGoogle Scholar
Shioya, T., Metric measure geometry: Gromov’s theory of convergence and concentration of metrics and measures, volume 25 of IRMA Lectures in Mathematics and Theoretical Physics, EMS Publishing House, Zürich, 2016.CrossRefGoogle Scholar
Sturm, K.-T., “Probability measures on metric spaces of nonpositive curvature” in P. Auscher, T. Coulhon and A. Grigoryan (eds.), Heat Kernels and Analysis on Manifolds, Graphs, and Metric Spaces (Paris, 2002), volume 338 of Contemporary Mathematics, American Mathematical Society, Providence, RI, 2003, pp. 357–390.Google Scholar
Villani, C., Optimal transport, volume 338 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], Springer-Verlag, Berlin, 2009. Old and new.CrossRefGoogle Scholar