Combinatorics and Arithmetic for Physics: special days

Collection Combinatorics and Arithmetic for Physics: special days

Organizer(s) Gérard H.E. Duchamp, Maxim Kontsevich, Gleb Koshevoy et Vincel Hoang Ngoc Minh
Date(s) 30/11/2021 - 02/12/2021
linked URL
00:00:00 / 00:00:00
18 20

Ambitropical Convexity, Mean Payoff Games and Nonarchimedean Convex Programming

By Stéphane Gaubert

Convex sets can be defined over ordered fields with a non-archimedean valuation. Then, tropical convex sets arise as images by the valuation of non-archimedean convex sets. The tropicalization of polyhedra and spectrahedra can be described in terms of deterministic and stochastic games with mean payoff, being characterized in terms of sub or super-fixed point sets of Shapley operators, which determine the value of the game. This is motivated by open complexity issues in linear programming. We shall discuss here especially a generalization of tropical convexity: considering fixed point sets of Shapley operators, instead of sub or super-fixed points sets, leads to a richer “ambitropical” theory, which includes tropical convexity and its dual in a unified framework. We shall present several characterizations of ambitropical convex sets, with features related to normed spaces (nonexpansive retracts and hyperconvexity), lattice theory (order preserving retracts), or of a combinatorial nature (cell decompositions in alcoved polyhedra).

The results on ambitropical convexity is from a work with Akian and Vannucci; the ones on the tropicalization of nonarchimedean convex sets are from works with Allamigeon, Benchimol, Joswig and Skomra.

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