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An Elementary Introduction to Modern Convex Geometry
 in Flavors of Geometry
, 1997
"... Introduction to Modern Convex Geometry KEITH BALL Contents Preface 1 Lecture 1. Basic Notions 2 Lecture 2. Spherical Sections of the Cube 8 Lecture 3. Fritz John's Theorem 13 Lecture 4. Volume Ratios and Spherical Sections of the Octahedron 19 Lecture 5. The BrunnMinkowski Inequality and It ..."
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Cited by 172 (2 self)
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Introduction to Modern Convex Geometry KEITH BALL Contents Preface 1 Lecture 1. Basic Notions 2 Lecture 2. Spherical Sections of the Cube 8 Lecture 3. Fritz John's Theorem 13 Lecture 4. Volume Ratios and Spherical Sections of the Octahedron 19 Lecture 5. The BrunnMinkowski Inequality
The Shapley Value on Convex Geometries
, 2000
"... A game on a convex geometry is a realvalued function defined on the family L of the closed sets of a closure operator which satisfies the finite MinkowskiKreinMilman property. If L is the boolean algebra 2 N then we obtain an nperson cooperative game. Faigle and Kern investigated games where L i ..."
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Cited by 29 (5 self)
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A game on a convex geometry is a realvalued function defined on the family L of the closed sets of a closure operator which satisfies the finite MinkowskiKreinMilman property. If L is the boolean algebra 2 N then we obtain an nperson cooperative game. Faigle and Kern investigated games where L
The core of games on convex geometries
, 1999
"... A game on a convex geometry is a realvalued function defined on the family L of the closed sets of a closure operator which satisfies the finite MinkowskiKreinMilman property. If L is the Boolean algebra 2^N then we obtain a nperson cooperative game. We will introduce convex and quasiconvex gam ..."
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Cited by 16 (3 self)
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A game on a convex geometry is a realvalued function defined on the family L of the closed sets of a closure operator which satisfies the finite MinkowskiKreinMilman property. If L is the Boolean algebra 2^N then we obtain a nperson cooperative game. We will introduce convex and quasiconvex
The Convex Geometry of Linear Inverse Problems
, 2010
"... In applications throughout science and engineering one is often faced with the challenge of solving an illposed inverse problem, where the number of available measurements is smaller than the dimension of the model to be estimated. However in many practical situations of interest, models are constr ..."
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Cited by 189 (20 self)
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are constrained structurally so that they only have a few degrees of freedom relative to their ambient dimension. This paper provides a general framework to convert notions of simplicity into convex penalty functions, resulting in convex optimization solutions to linear, underdetermined inverse problems
Scalable Frames and Convex Geometry
"... Abstract. The recently introduced and characterized scalable frames can be considered as those frames which allow for perfect preconditioning in the sense that the frame vectors can be rescaled to yield a tight frame. In this paper we define mscalability, a refinement of scalability based on the nu ..."
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Cited by 2 (0 self)
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on the number of nonzero weights used in the rescaling process, and study the connection between this notion and elements from convex geometry. Finally, we provide results on the topology of scalable frames. In particular, we prove that the set of scalable frames with “small ” redundancy is nowhere dense
A CLASS OF INFINITE CONVEX GEOMETRIES
"... Abstract. Various characterizations of finite convex geometries are well known. This note provides similar characterizations for possibly infinite convex geometries whose lattice of closed sets is strongly coatomic and lower continuous. Some classes of examples of such convex geometries are given. ..."
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Abstract. Various characterizations of finite convex geometries are well known. This note provides similar characterizations for possibly infinite convex geometries whose lattice of closed sets is strongly coatomic and lower continuous. Some classes of examples of such convex geometries are given.
Some Properties of the Core on Convex Geometries
"... A game on a convex geometry was introduced by Bilbao as a model of partial cooperation. We investigate some properties of the core of a game on a convex geometry. ..."
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A game on a convex geometry was introduced by Bilbao as a model of partial cooperation. We investigate some properties of the core of a game on a convex geometry.
Marginal Operators for Games on Convex Geometries
"... We introduce marginal worth vectors and quasisupermodular games on convex geometries. Furthermore, we study some properties of the minimal marginal operator and the maximal marginal operator on the space of the games on convex geometries. ..."
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We introduce marginal worth vectors and quasisupermodular games on convex geometries. Furthermore, we study some properties of the minimal marginal operator and the maximal marginal operator on the space of the games on convex geometries.
Optimum basis of finite convex geometry
, 2014
"... Convex geometries form a subclass of closure systems with unique criticals, or UCsystems. We show that the Fbasis introduced in [6] for UCsystems, becomes optimum in convex geometries, in two essential parts of the basis. The last part of the basis can be optimized, when the convex geometry eit ..."
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Cited by 2 (1 self)
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Convex geometries form a subclass of closure systems with unique criticals, or UCsystems. We show that the Fbasis introduced in [6] for UCsystems, becomes optimum in convex geometries, in two essential parts of the basis. The last part of the basis can be optimized, when the convex geometry
An incidence Hopf Algebra of Convex Geometries
, 2011
"... A lattice L is “meetdistributive” if for each element of L, the meets of the elements directly below it form a Boolean lattice. These objects are in bijection with “convex geometries”, which are an abstract model of convexity. Do they give rise to an incidence Hopf algebra of convex geometries? ..."
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A lattice L is “meetdistributive” if for each element of L, the meets of the elements directly below it form a Boolean lattice. These objects are in bijection with “convex geometries”, which are an abstract model of convexity. Do they give rise to an incidence Hopf algebra of convex geometries?
Results 1  10
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