Tropical convexity, mean payoff games and nonarchimedean convex programming
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Tropical convexity, mean payoff games and nonarchimedean convex programming

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Tropical Convexity, Mean Payoff Games and Nonarchimedean Convex Programming

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CC Attribution 3.0 Unported:
You are free to use, adapt and copy, distribute and transmit the work or content in adapted or unchanged form for any legal purpose as long as the work is attributed to the author in the manner specified by the author or licensor. 
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2021

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English

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Abstract 
Convex sets can be defined over ordered fields with a nonarchimedean valuation. Then, tropical convex sets arise as images by the valuation of nonarchimedean convex sets. The tropicalization of polyhedra and spectrahedra are of special in terest, since they can be described in terms of deterministic and stochastic games with mean payoff. In that way, one gets a correspondence between classes of zero sum games, with an unsettled complexity, and classes of semilagebraic convex op timization problems over nonarchimedean fields. We shall discuss applications of this correspondence, including a counter example concerning the complexity of interior point methods, and the fact that nonarchimedean spectrahedra have precisely the same images by the valuation as convex semialgebraic sets. This is based on works with Allamigeon, Benchimol, Joswig and Skomra.

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Geometry
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