Joint Seminar on Gromov-Hausdorff distance geometry – 2025

On this web page we post materials for joint seminar devoted to the geometry of the Gromov-Hausdorff distance.

Henry Adams

Alexey Tuzhilin

Alexander Ivanov

Anton Vikhrov

Ivan Mikhailov

Boris Nesterov

Our meetings

• 1-st meeting held on Tuesday, March 4, 2025, at 9am EST = 5pm MSK
• 2-nd meeting held on Tuesday, April 1, 2025, at 9am EST = 5pm MSK
• Forthcoming 3-nd meeting is scheduled on Tuesday, April 29, 2025, at 9am EST = 5pm MSK

Presentations of the talks on the seminar

• Alexey Tuzhilin. Introduction.
• Anton Vikhrov. Geodesics in Gromov-Hausdorff proper class.
• Boris Nesterov. Gromov–Hausdorff distance between clouds of special type. Calculating distance to cloud of bounded metric spaces.

Some problems

• Geodesics in GH-class. Is it true that the GH-distance is geodesic? Which geodesic can be extended beyond some of its end?
• Investigate isometries of GH-class, global and local. Is it true that the Hausdorff mapping that takes a metric spaces to its hyperspace (the space of all closed bounded subsets, endowed with Hausdorff metric) is isometric? Are there isometric clouds?
• Calculate the GH- and Lipschitz distances between clouds.
• Which metric spaces can be isometrically embedded into GH-space (of compact metric spaces)? Is it true that each unbounded metric space can be isometrically embedded into GH-class?
• Continue investigation of GH-distances to spaces with one non-zero distance.
• Continue investigation of GH-distances between spheres (in Euclidean spaces, and, more general, in normed spaces).
• Calculate or estimate the GH-distances between Euclidean balls.
• Calculate or estimate the GH-distances between metric spaces from other natural families.
• When the Steiner problem of existence of a shortest tree has solutions in GH-class?
• Application of GH-distance to graph theory and some problems like Borsuk conjecture.
• Application of algebraic geometry to calculation and estimation of GH-distance.
• Investigate geometry GH-space and GH-space, e.g., are the spheres connected, are the balls convex, is it true that each ball center is uniquely determined, etc.?
• Investigate various approximations of the GH-distance, e.g., when the approximation can be found in polynomial time?

References

• Adams H., Frick F., Majhi S., McBride N. Hausdorff vs Gromov–Hausdorff distances, 2025, ArXiv e-prints, arXiv:2309.16648v5 [math.MG]
• Adams H., Majhi S., Manin F., Virk Z., Zava N. Lower-bounding the Gromov–Hausdorff distance in metric graphs, 2024, ArXiv e-prints, arXiv:2411.09182 [math.MG]
• Bogataya S.I., Bogatyy S.A., Redkozubov V.V., Tuzhilin A.A. Clouds in Gromov-Hausdorff Class: their completeness and centers, 2022, ArXiv e-prints, arXiv:2202.07337v1 [math.MG]
• Borzov S.I., Ivanov A.O., Tuzhilin A.A. Extendability of Metric Segments in Gromov–Hausdorff Distance, 2020, arXiv:2009.00458v1 [math.MG]
• Grigor’ev D.S., Ivanov A.O., Tuzhilin A.A. Gromov–Hausdorff Distance to Simplexes, 2019, ArXiv e-prints, arXiv:1906.09644v1 [math.MG]
• Harrison M., Jeffs R.A. Quantitative upper bounds on the Gromov–Hausdorff distance between spheres, 2024, ArXiv e-prints, arXiv:2309.11237v3 [math.MG].
• Ivanov A.O., Lychagina E.S., Tuzhilin A.A. Metric Space Recognition by Gromov–Hausdorff Distances to Simplexes, 2024, ArXiv e-prints, arXiv:2412.18949v1 [math.MG]
• Ivanov A.O., Mikhailov I.N., Tuzhilin A.A. Gromov-Hausdorff Geometry of Metric Trees, 2024, ArXiv e-prints, arXiv:2412.18888v1 [math.MG]
• Ivanov A.O., Tsvetnikov R.A., Tuzhilin A.A. Path Connectivity of Spheres in the Gromov-Hausdorff Class, 2021, ArXiv e-prints, arXiv:2111.06709 [math.MG]
• Ivanov A.O., Tuzhilin A.A. Geometry of Compact Metric Space in Terms of Gromov-Hausdorff Distances to Regular Simplexes, 2016, ArXiv e-prints, arXiv:1607.06655v1 [math.MG]
• Ivanov A.O., Tuzhilin A.A. Gromov–Hausdorff Distance, Irreducible Correspondences, Steiner Problem, and Minimal Fillings, 2016, ArXiv e-prints, arXiv:1604.06116v1 [math.MG]
• Ivanov A.O., Tuzhilin A.A. Solution to Generalized Borsuk Problem in Terms of the Gromov–Hausdorff Distances to Simplexes. 2019, ArXiv e-prints, arXiv:1906.10574 [math.MG]
• Ivanov A.O., Tuzhilin A.A. The Gromov–Hausdorff Distance between Simplexes and Two-Distance Spaces, 2019, ArXiv e-prints, arXiv:1907.09942v1 [math.MG]
• Ivanov A.O., Tuzhilin A.A. The Gromov-Hausdorff Distances between Simplexes and Ultrametric Spaces, 2019, ArXiv e-prints, arXiv:1907.03828v1 [math.MG]
• Ji Y., Tuzhilin A.A. Gromov-Hausdorff Distance Between Segment and Circle, 2021, ArXiv e-prints, arXiv:2101.05762 [math.MG]
• Martin S.R. Gromov-Hausdorff distances from simply connected geodesic spaces to the circle , 2024, ArXiv e-prints, arXiv:2404.05153v2 [math.MG]
• Martin S.R. Some novel constructions of Gromov-Hausdorff-optimal correspondences between spheres , 2025, ArXiv e-prints, arXiv:2409.02248v2 [math.MG]
• Memoli F., Smith Z.T. Embedding-projection correspondences for the estimation of the Gromov-Hausdorff distance, 2024, arXiv preprint arXiv:2407.03295 [math.MG]
• Mikhailov I.N., Tuzhilin A.A. When the Gromov-Hausdorff distance between finite-dimensional space and its subset is finite?, ArXiv e-prints, arXiv:2411.1353 [math.MG]
• Mikhailov I.N. Gromov-Hausdorff distances between normed spaces, ArXiv e-prints, arXiv:2407.01388 [math.MG]
• Mikhailov I.N. Ultrametric spaces and clouds, ArXiv e-prints, arXiv:2501.19346v1 [math.MG]
• Talipov T. Gromov-Hausdorff distance between vertex sets of regular polygons inscribed in a given circle, 2022, ArXiv e-prints, arXiv:2210.09971v1 [math.MG]
• Tuzhilin A.A. Calculation of Minimum Spanning Tree Edges Lengths using Gromov–Hausdorff Distance, 2016, ArXiv e-prints, arXiv:1605.01566v1 [math.MG]
• Vikhrov A.A. Density of generic metric spaces in the Gromov-Hausdorff class, 2023, ArXiv e-prints, arXiv:2302.12865 [math.MG]