Point configurations and convex polytopes play central roles in computational geometry and discrete geometry. For many problems, their combinatorial structures, i.e., the underlying oriented matroids up to isomorphism, are often more important than their metric structures. For example, the convexity, the face lattice of the convex hull and all possible triangulations of a given point configurat...

It is well known that every closure system can be represented by an implicational base, or the set of its meet-irreducible elements. In Horn logic, these are respectively as expressions and characteristic models. this paper, we consider problem translating between two representations in acyclic convex geometries. Quite surprisingly, show context already harder than dualization distributive latt...

As the basic issues of computational geometry, intersection and union of convex polygons can solve lots of problems, such as economy and military affairs. And privacy-preserving judgment of the intersection and union for convex polygons are most popular issues for information security. Traditional method of making the polygons public does not satisfy the requirements of personal privacy. In thi...

The partial KKM principle for an abstract convex space is an abstract form of the classical KKM theorem. In this paper, we derive generalized forms of the Ky Fan minimax inequality, the von Neumann-Sion minimax theorem, the von Neumann-Fan intersection theorem, the Fan-type analytic alternative, and the Nash equilibrium theorem for abstract convex spaces satisfying the partial KKM principle. Th...

By using a fixed-point theorem in G-convex spaces due to the first author, an existence result for abstract nonlinear inequalities without any monotonicity assumptions is established. As a consequence of our result, we obtain some further generalizations of recent known results. As application, an existence theorem for perturbed saddle point problems is obtained in noncompact G-convex spaces. @...

In this paper, we introduce and study a class of generalized symmetric vector quasi-equilibrium problems in abstract convex spaces. By virtue of the properties of Γ-convex and KC-map, we give some sufficient conditions to guarantee the existence of solutions for the generalized symmetric vector quasi-equilibrium problems in abstract convex spaces. As application, we show an existence theorem of...