Another inverse of the Berge maximum theorem (Q2720306)

By using an idea from \textit{H. Komiya} [Econ. Theory 9, No. 2, 371-375 (1997; Zbl 0872.90016)], the authors obtain an inverse of the famous Berge theorem, by proving the following result:NEWLINENEWLINENEWLINELet \(X\) be a topological vector space, and \(Y\) a metric topological vector space whose balls are convex, and \(\Gamma: X\multimap Y\) a \(\sigma\)-selectionable map. Then there exists a continuous function \(f: X\times Y\to [0,1]\) such thatNEWLINENEWLINENEWLINE(i) \(\Gamma(x)= \{y\in Y: f(x,y)= \max_{z\in Y} f(x,z)\}\) for any \(x\in X\); andNEWLINENEWLINENEWLINE(ii) \(f(x,y)\) is quasi-concave in \(y\) for any \(x\in X\).NEWLINENEWLINENEWLINEBy deducing fixed point theorem, the authors then show that the Kakutani, as well as Schauder-fixed point theorems are each equivalent to the Brouwer fixed point theorem.