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We show that for every d-dimensional polytope, the hypergraph whose nodes are kfaces and whose hyperedges are (k +1)-faces of the polytope is strongly (d −k)-vertex connected, for each 0 ≤ k ≤ d − 1. 

Abstract We show that the tropicalization of an irreducibled-dimensional variety over a field of characteristic 0 is$$(d-\ell )$$ $(d-\ell )$-connected through codimension one, where$$\ell $$ $\ell$is the dimension of the lineality space of the tropicalization. From this we obtain a higher connectivity result for skeleta of rational polytopes. We also prove a tropical analogue of the Bertini Theorem: the intersection of the tropicalization of an irreducible variety with a generic hyperplane is again the tropicalization of an irreducible variety. 

Abstract Let $$K$$ be a real closed field with a nontrivial non-archimedean absolute value. We study a refined version of the tropicalization map, which we call real tropicalization map, that takes into account the signs on $$K$$. We study images of semialgebraic subsets of $K^n$ under this map from a general point of view. For a semialgebraic set $$S \subseteq K^n$$ we define a space $$S_r^{{\operatorname{an}}}$$ called the real analytification, which we show to be homeomorphic to the inverse limit of all real tropicalizations of $$S$$. We prove a real analogue of the tropical fundamental theorem and show that the tropicalization of any semialgebraic set is described by tropicalization of finitely many inequalities, which are valid on the semialgebraic set. We also study the topological properties of real analytification and tropicalization. If $$X$$ is an algebraic variety, we show that $$X_r^{{\operatorname{an}}}$$ can be canonically embedded into the real spectrum $$X_r$$ of $$X$$, and we study its relation with the Berkovich analytification of $$X$$.