Existence of solutions and star-shapedness in Minty variational inequalities

Crespi Giovanni (), Ginchev Ivan () and Rocca Matteo ()
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Crespi Giovanni: Department of Economics, University of Insubria, Italy
Ginchev Ivan: Department of Mathematics, Technical University of Varna, Bulgaria
Rocca Matteo: Department of Economics, University of Insubria, Italy

Economics and Quantitative Methods from Department of Economics, University of Insubria

Abstract: Minty variational inequalities have proven to define a stronger notion of equilibrium than Stampacchia variational inequalities. This conclusion leads to argue that some regularity, e.g. convexity or generalized convexity, has to be implicit for any function that admits a solution of the corresponding integrable Minty variational inequality. Quasi-convexity arises almost naturally when functions of one variable are involved. However some differences appear when considering functions of several variables. In this case we show that existence of a solution does not necessarily imply quasi-convexity of the function and instead we prove that the level sets of the function must be star-shaped at a point which is a solution of the Minty variational inequality.

Keywords: Minty variational inequality; generalized convexity; star-shaped sets; existence of solutions (search for similar items in EconPapers)
JEL-codes: C60 C61 (search for similar items in EconPapers)
Pages: 7 pages
Date: 2002-07
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