Isometric structure of transportation cost spaces on finite metric spaces

The paper is devoted to isometric Banach-space-theoretical structure of transportation cost (TC) spaces on finite metric spaces. The TC spaces are also known as Arens-Eells, Lipschitzfree, or Wasserstein spaces. A new notion of a roadmap pertinent to a transportation problem on a finite metric space has been introduced and used to simplify proofs for the results on representation of TC spaces as quotients of l1 spaces on the edge set over the cycle space. A Tolstoi-type theorem for roadmaps is proved, and directed subgraphs of the canonical graphs, which are supports of maximal optimal roadmaps, are characterized. Possible obstacles for a TC space on a finite metric space X preventing them from containing subspaces isometric to ln ∞ have been found in terms of the canonical graph of X . The fact that TC spaces on diamond graphs do not contain l ∞ isometrically has been derived. In addition, a short overview of known results on the isometric structure of TC spaces on finite metric spaces is presented. Acknowledgement: The first-named author gratefully acknowledges the support by Atilim University. This paper was written while the first-named author was on research leave supported by Atilim University. The second-named author gratefully acknowledges the support by the National Science Foundation grant NSF DMS-1953773. 1 Basic definitions and results The theory of transportation cost spaces launched by Kantorovich and Gavurin [Kan42, KG49] initially had been developed as a study of special norms pertaining to function spaces on finite metric spaces. However, its further progress turned into the direction of infinite and continuous setting rather than the discrete one in papers of Kantorovich and Rubinstein [Kan42, Kan48, KR57, KR58], and this stream has become dominant. See [ABS21, AGS08, FG21, Gar18, KA82, Vil03, Vil09, Wea18]. Nevertheless, researches within Theoretical Computer Science on the transportation cost, which computer scientists renamed to earth mover distance [RTG98], along with some of the recent works in metric geometry and Banach space theory focus on the case of finite metric spaces bringing this area back into spotlight. See, for example, [AFGZ21, BMSZ20+, Cha02, DKO20, DKO21, IT03, KKMR09, KMO20, KN06, Nao21, NR17, NS07, RTG98]. More details on the history of the subject can be found in [Vil09, Chapter 3] and [OO19, Section 1.6]. This work deals with the isometric theory of transportation cost spaces on finite metric spaces. It has to be pointed out that the transportation cost spaces have ∗Corresponding author, ostrovsm@stjohns.edu

[1]  Some natural subspaces and quotient spaces of $L^1$ , 2017, 1702.06049.

[2]  Marek C'uth,et al.  Isometric embedding of ℓ₁ into Lipschitz-free spaces and ℓ_{∞} into their duals , 2016, 1604.04131.

[3]  M. Ostrovskii,et al.  Generalized Transportation Cost Spaces , 2019, Mediterranean Journal of Mathematics.

[4]  L. Ambrosio,et al.  Lectures on Optimal Transport , 2021, UNITEXT.

[5]  A. Zvavitch,et al.  Geometry and volume product of finite dimensional Lipschitz-free spaces , 2019, Journal of Functional Analysis.

[6]  Gershon Wolansky,et al.  Optimal Transport , 2021 .

[7]  Assaf Naor,et al.  Metric dimension reduction: A snapshot of the Ribe program , 2018, Proceedings of the International Congress of Mathematicians (ICM 2018).

[8]  Alexandre Godard.  Tree metrics and their Lipschitz-free spaces , 2009, 0904.3178.

[9]  M. R. Rao,et al.  Combinatorial Optimization , 1992, NATO ASI Series.

[10]  R. Mortini,et al.  Lipschitz algebras , 2021, Extension Problems and Stable Ranks.

[11]  Hyunjoong Kim,et al.  Functional Analysis I , 2017 .

[12]  Gideon Schechtman,et al.  Planar Earthmover is not in L_1 , 2005, 2006 47th Annual IEEE Symposium on Foundations of Computer Science (FOCS'06).

[13]  Prasad Raghavendra,et al.  Coarse Differentiation and Multi-flows in Planar Graphs , 2007, APPROX-RANDOM.

[14]  Marek C'uth,et al.  ISOMETRIC REPRESENTATION OF LIPSCHITZ-FREE SPACES OVER CONVEX DOMAINS IN FINITE-DIMENSIONAL SPACES , 2016, 1610.03966.

[15]  M. Ostrovskii.  Metric Embeddings: Bilipschitz and Coarse Embeddings into Banach Spaces , 2013 .

[16]  S. Dilworth,et al.  Lipschitz-free Spaces on Finite Metric Spaces , 2018, Canadian Journal of Mathematics.

[17]  A. Naor.  Impossibility of almost extension , 2020, 2009.11373.

[18]  Moses Charikar,et al.  Similarity estimation techniques from rounding algorithms , 2002, STOC '02.

[19]  D. J. H. Garling,et al.  Analysis on Polish Spaces and an Introduction to Optimal Transportation , 2017 .

[20]  Aranyak Mehta,et al.  On earthmover distance, metric labeling, and 0-extension , 2006, STOC '06.

[21]  L. Ambrosio,et al.  Gradient Flows: In Metric Spaces and in the Space of Probability Measures , 2005 .

[22]  A. Figalli,et al.  An Invitation to Optimal Transport, Wasserstein Distances, and Gradient Flows , 2021 .

[23]  C. Villani.  Topics in Optimal Transportation , 2003 .

[24]  Leonidas J. Guibas,et al.  A metric for distributions with applications to image databases , 1998, Sixth International Conference on Computer Vision (IEEE Cat. No.98CH36271).

[25]  Yuval Rabani,et al.  On Lipschitz extension from finite subsets , 2015, 1506.04398.

[26]  Subhash Khot,et al.  Nonembeddability theorems via Fourier analysis , 2005, 46th Annual IEEE Symposium on Foundations of Computer Science (FOCS'05).

[27]  J. Bourgain.  The metrical interpretation of superreflexivity in banach spaces , 1986 .

[28]  Anupam Gupta,et al.  Cuts, Trees and ℓ1-Embeddings of Graphs* , 1999, 40th Annual Symposium on Foundations of Computer Science (Cat. No.99CB37039).

[29]  H. Whitney.  2-Isomorphic Graphs , 1933 .

[31]  James G. Oxley,et al.  Matroid theory , 1992 .

[32]  G. Pisier,et al.  Séries de variables aléatoires vectorielles indépendantes et propriétés géométriques des espaces de Banach , 1976 .