Abstract: (Note: 3:30-4 p.m. will be an open/informal talk with the speaker) We consider (finitely supported) transportation problems on a metric space M. They form a vector space TP(M). The optimal transportation cost for such transportation problems is a norm on this space. Such normed spaces were introduced and studied by Kantorovich and his students in 1940-1950s years. This development lead to terms: Kantorovich distance, Kantorovich-Rubinstein distance, and transportation cost. Simultaneously, another group of researchers, starting with Markov (1941) started the study of algebraically “free” topological (or metric) structures which contain a given topological (or metric) structure as a substructure. On these lines Arens and Eells (1956) constructed the space which coincides with the space of transportation problems with the norm equal to the optimal transportation cost. In this connection you can meet such terms as Arens-Eells space and Lipschitz-free space. I am going to talk about geometry of such normed spaces. My results presented in this talk, mentioned in it, or related to it, can be found in joint papers with Stephen Dilworth, Seychelle Khan, Denka Kutzarova, Mutasim Mim, and Sofiya Ostrovska (available on arXiv).
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