×

Non-Archimedean transportation problems and Kantorovich ultra-norms. (English) Zbl 1353.90027

Summary: We study a non-archimedean (NA) version of transportation problems and introduce naturally arising ultra-norms which we call Kantorovich ultra-norms. For every ultra-metric space and every NA valued field (e.g., the field \(\mathbb{Q}_p\) of \(p\)-adic numbers) the naturally defined inf-max cost formula achieves its infimum. We also present NA versions of the Arens-Eells construction and of the integer value property. We introduce and study free NA locally convex spaces. In particular, we provide conditions under which these spaces are normable by Kantorovich ultra-norms and also conditions which yield NA versions of Tkachenko-Uspenskij theorem about free abelian topological groups.

MSC:

90B06 Transportation, logistics and supply chain management
54E35 Metric spaces, metrizability
54H11 Topological groups (topological aspects)
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] R. Arens and J. Eells, “On embedding uniform and topological spaces,” Pacific J. Math. 6, 397-403 (1956). · Zbl 0073.39601 · doi:10.2140/pjm.1956.6.397
[2] A. Arhangel’skii and M. Tkachenko, Topological Groups and Related Structures, Atlantis Studies in Math. Series 1, Editor J. van Mill (Atlantis Press, World Sci., Amsterdam-Paris, 2008). · Zbl 1323.22001 · doi:10.2991/978-94-91216-35-0
[3] R. Burkard, M. DellAmico and S. Martello, Assignment Problems (SIAM, Philadelphia, 2009). · Zbl 1196.90002 · doi:10.1137/1.9780898717754
[4] Y. Deng and W. Du, “The Kantorovich metric in computer science:A brief survey,” Elect. Notes Theor. Comp. Sci. 253 (3), 73-82 (2009). · doi:10.1016/j.entcs.2009.10.006
[5] J. Flood, Free Topological Vector Spaces, DissertationesMath. CCXXI (PWN,Warczawa, 1984). · Zbl 0545.46052
[6] S. Gao and V. Pestov, “On a universality property of some abelian Polish groups,” Fund.Math. 179 (1), 1-15 (2003). · Zbl 1057.22003 · doi:10.4064/fm179-1-1
[7] L. V. Kantorovich, “On the transfer of masses,” Dokl. Akad. Nauk USSR 37 (7-8), 227-229 (1942) [in Russian].
[8] S. V. Lyudkovskii, “Non-Archimedean free Banach space,” Fundam. Prikl.Mat. 1:4, 979-987 (1995). · Zbl 0872.46038
[9] M. Megrelishvili and M. Shlossberg, “Notes on non-archimedean topological groups,” Top. Appl. 159, 2497-2505 (2012). · Zbl 1247.22002 · doi:10.1016/j.topol.2011.06.069
[10] M. Megrelishvili and M. Shlossberg, “Free non-archimedean topological groups,” Comment. Math. Univ. Carolin. 54:2, 273-312 (2013). · Zbl 1289.54105
[11] C. Perez-Garcia and W. H. Schikhof, Locally Convex Spaces over Non-Archimedean Valued Fields (Cambridge Univ. Press, New York, 2010). · Zbl 1193.46001 · doi:10.1017/CBO9780511729959
[12] J. Melleray, F. V. Petrov and A.M. Vershik, “Linearly rigid metric spaces and the embedding problem,” Fund. Math. 99 (2), 177-194 (2008). · Zbl 1178.46016 · doi:10.4064/fm199-2-6
[13] V. Pestov, “Topological groups: where to from here?,” Topology Proc. 24, 421-502 (1999). · Zbl 1026.22002
[14] S. T. Rachev and L. Rüschendorf, Mass Transportation Problems, Vol. I: Theory, Vol. II: Applications (Springer, New York-Berlin-Heidelberg, 1998). · Zbl 0990.60500
[15] D. A. Raikov, “Free locally convex spaces for uniform spaces,” Mat. Sb. (N.S.) 63 (105), 582-590 (1964) [Russian]. · Zbl 0117.39902
[16] A. C. M. van Rooij, Non-Archimedean Functional Analysis, Monographs and Textbooks in Pure and AppliedMath. 51 (Marcel Dekker, Inc., New York, 1978). · Zbl 0396.46061
[17] M. Sakarovitch, Linear Programming, Springer Texts in Electrical Engineering (Springer-Verlag, New York, 1983). · Zbl 0521.90070
[18] P. Schneider, Nonarchimedean Functional Analysis (Berlin, Springer, 2002). · Zbl 0998.46044 · doi:10.1007/978-3-662-04728-6
[19] K. Shamseddine, “On the topological structure of the Levi-Civita field,” J. Math. Anal. Appl. 368, 281-292 (2010). · Zbl 1190.26036 · doi:10.1016/j.jmaa.2010.02.018
[20] O. V. Sipacheva, “The topology of free topological groups,” J.Math. Sci. (N.Y.) 131 (4), 5765-5838 (2005). · Zbl 1068.22002 · doi:10.1007/s10958-005-0445-z
[21] M.G. Tkachenko, “On completeness of free abelian topological groups,” Soviet Math. Dokl. 27, 341-345 (1983). · Zbl 0521.22002
[22] V. V. Uspenskij, “Free topological groups of metrizable spaces,” Math. USSR Izvestiya 37, 657-680 (1991). · Zbl 0739.22002 · doi:10.1070/IM1991v037n03ABEH002163
[23] A.M. Vershik, “The Kantorovichmetric: the initial history and little-known applications,” J.Math. Sci. (N.Y.) 133 (4), 1410-1417 (2006). · Zbl 1090.28009 · doi:10.1007/s10958-006-0056-3
[24] C. Villani, Optimal Transport, Old and New, SCSM338 (Springer, 2008). · Zbl 1156.53003
[25] N. Weaver, Lipschitz Algebras (World Sci. Publ. Co., Inc., River Edge, NJ, 1999). · Zbl 0936.46002 · doi:10.1142/4100
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.