Equilibrium Theorem as the consequence of the Steinhaus Chessboard Theorem by Marian Turzanski

Topology Proceedings 25 Summer (2000), pp. 645-653

Equilibrium Theorem as the consequence of the Steinhaus Chessboard Theorem

Marian Turzanski

Kulpa proved the existence of a stable-like point (Equilibrium Theorem) and applied this theorem to show the existence of rational divisions of bounded Lebegue measurable sets in Euclidean spaces. We present an algorithm for determining on the Euclidean plane the place where the equilibrium points are. For this purpose, we use the Steinhaus chessboard theorem. The existence of market equilibrium is a classical problem in economics (Walras, von Neumann, Nash). The Brouwer fixed point theorem was the main mathematical tool in Nash's paper, for which he won the Nobel prize in economics. The Brouwer Theorem is an easy consequence of Kulpa's Equilibrium Theorem. Hence, an algorithm for determining a fixed point is also given.

Keywords: The Poincare acute theorem, the Brouwer fixed point theorem, the Steinhaus chessboard theorem, the Kulpa's equilibrium theorem
Mathematics Subject Classification: 54H25 54F45 54C20 54C35

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Topology Proceedings, Volume 25 Summer (2000)
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