Tropical tensors arise naturally in many areas, such as game theory, optimal control, and approximation theory. However, very little is known about their properties. In this talk, we first introduce inner and outer approximations of a tropical polytope, which can help us study tropical tensors. As corollaries, we obtain interesting properties of tropical linear regressions and best rank-one approximations of tropical tensors, which for example are very useful in inferring hidden information from equilibria in repeated invitations to tenders. This talk is based on a joint work with Marianne Akian, Stéphane Gaubert, and Omar Saadi.
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