We propose a notion of competitive equilibrium in an abstract setting called a Convex Economy using a concept of convexity borrowed from Convex Geometry. The "magic” of linear equilibrium prices is put into perspective in this abstract setting. The abstract notion of competitive equilibrium is applied to a variety of convex economies and versions of the first and second fundamental welfare theo...

We introduce a notion of a broken circuit and an NBC complex for an (abstract) convex geometry. Based on these definitions, we shall show the analogues of the Whitney-Rota’s formula and Brylawski’s decomposition theorem for broken circuit complexes on matroids for convex geometries. We also present an Orlik-Solomon type algebra on a convex geometry, and show the NBC generating theorem. This not...

Convex hull of some given points is the intersection of all convex sets containing them. It is used as primary structure in many other problems in computational geometry and other areas like image processing, model identification, geographical data systems, and triangular computation of a set of points and so on. Computing the convex hull of a set of point is one of the most fundamental and imp...

We propose a new abstract definition of equilibrium in the spirit of competitive equilibrium: a profile of alternatives and a public ordering (expressing prestige, price or a social norm), such that each agent prefers his assigned alternative to all lower-ranked ones. The equilibrium operates in an abstract setting built upon a concept of convexity borrowed from Convex Geometry. We apply the co...

We propose a new abstract definition of competitive equilibrium: a profile of alternatives and a public ordering (prestige, price or a social norm) such that each agent prefers his assigned alternative to all lower-ranked ones. The equilibrium operates in an abstract setting built upon a concept of convexity borrowed from Convex Geometry. The "magic" of linear equilibrium prices is put into per...

Weidong Min Abstract. This paper presents an algorithm based on shrunken polyhedron and mapping to mesh a convex polyhedron using mixed elements, with mostly hexahedra, but including some tetrahedra, pyramids, and wedges. It is part of a scheme for automatically meshing an arbitrary 3D geometry. The scheme rst decomposes a 3D geometry into a set of convex polyhedra. Then the boundary quadrilate...

Karoly Bezdek and Egon Schulte are two geometers who have made important and numerous contributions to Discrete Geometry. Both have trained many young mathematicians and have actively encouraged the development of the area by professors of various countries. Karoly Bezdek studied the illumination problem among many other problems about packing and covering of convex bodies. More recently he has...