Steiner problems in optimal transport

Dahl, Jonathan

doi:10.1090/S0002-9947-2010-05065-2

Steiner problems in optimal transport
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by Jonathan Dahl

Trans. Amer. Math. Soc. 363 (2011), 1805-1819

DOI: https://doi.org/10.1090/S0002-9947-2010-05065-2

Published electronically: November 17, 2010

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Abstract:

We study the Steiner problem of finding a minimal spanning network in the setting of a space of probability measures with metric defined by the cost of optimal transport between measures. The existence of a solution is shown for the Wasserstein space $P_p(\mathcal {X})$ over any base space $\mathcal {X}$ which is a separable, locally compact Hadamard space. Structural results are given for the case $P_2(\mathbb {R}^n)$.

References

Luigi Ambrosio and Paolo Tilli, Topics on analysis in metric spaces, Oxford Lecture Series in Mathematics and its Applications, vol. 25, Oxford University Press, Oxford, 2004. MR 2039660
Martin R. Bridson and André Haefliger, Metric spaces of non-positive curvature, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 319, Springer-Verlag, Berlin, 1999. MR 1744486, DOI 10.1007/978-3-662-12494-9
Dmitri Burago, Yuri Burago, and Sergei Ivanov, A course in metric geometry, Graduate Studies in Mathematics, vol. 33, American Mathematical Society, Providence, RI, 2001. MR 1835418, DOI 10.1090/gsm/033
Manfredo Perdigão do Carmo, Riemannian geometry, Mathematics: Theory & Applications, Birkhäuser Boston, Inc., Boston, MA, 1992. Translated from the second Portuguese edition by Francis Flaherty. MR 1138207, DOI 10.1007/978-1-4757-2201-7
Stephanie Halbeisen, On tangent cones of Alexandrov spaces with curvature bounded below, Manuscripta Math. 103 (2000), no. 2, 169–182. MR 1796313, DOI 10.1007/s002290070018
Alexandr O. Ivanov and Alexei A. Tuzhilin, Minimal networks, CRC Press, Boca Raton, FL, 1994. The Steiner problem and its generalizations. MR 1271779
L. V. Kantorovich, On a problem of Monge, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 312 (2004), no. Teor. Predst. Din. Sist. Komb. i Algoritm. Metody. 11, 15–16 (Russian); English transl., J. Math. Sci. (N.Y.) 133 (2006), no. 4, 1383. MR 2117877, DOI 10.1007/s10958-006-0050-9
L. V. Kantorovich, On mass transportation, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 312 (2004), no. Teor. Predst. Din. Sist. Komb. i Algoritm. Metody. 11, 11–14 (Russian); English transl., J. Math. Sci. (N.Y.) 133 (2006), no. 4, 1381–1382. MR 2117876, DOI 10.1007/s10958-006-0049-2
John Lott and Cédric Villani, Ricci curvature for metric-measure spaces via optimal transport, Ann. of Math. (2) 169 (2009), no. 3, 903–991. MR 2480619, DOI 10.4007/annals.2009.169.903
Chikako Mese and Sumio Yamada, The parameterized Steiner problem and the singular plateau Problem via energy, Trans. Amer. Math. Soc. 358 (2006), no. 7, 2875–2895. MR 2216250, DOI 10.1090/S0002-9947-06-04089-X
Yu. V. Prokhorov, Convergence of random processes and limit theorems in probability theory, Teor. Veroyatnost. i Primenen. 1 (1956), 177–238 (Russian, with English summary). MR 0084896
Cédric Villani, Optimal transport, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 338, Springer-Verlag, Berlin, 2009. Old and new. MR 2459454, DOI 10.1007/978-3-540-71050-9