On the polytope of non-additive measures

IEEE Transactions on Fuzzy Systems, 2019

In this paper we study the polytope of 2-additive measures, an important subpolytope of the polytope of fuzzy measures. For this polytope, we obtain its combinatorial structure, namely the adjacency structure and the structure of 2-dimensional faces, 3-dimensional faces, and so on. Basing on this information, we build a triangulation of this polytope satisfying that all simplices in the triangulation have the same volume. As a consequence, this allows a very simple and appealing way to generate points in a random way in this polytope, an interesting problem arising in the practical identification of 2-additive measures. Finally, we also derive the volume, the centroid, and some properties concerning the adjacency graph of this polytope.

In this paper, a graph is assigned to any probability measure on the í µí¼Ž-algebra of Borel sets of a topological space. Using this construction, it is proved that given any number í µí±› (finite or infinite) there exists a nonregular graph such that its clique, chromatic, and dominating number equals í µí±›.

Eusflat, 2009

Given a capacity, the set of dominating k-additive capacities is a convex polytope; thus, it is defined by its vertices. In this paper we deal with the problem of deriving a procedure to obtain such vertices in the line of the results of Shapley and Ichiishi for the additive case. We propose an algorithm to determine the vertices of the k-additive monotone core. Then, we characterize the vertices of the n-additive core and finally, we explore the possible translations for the k-additive case.

Journal of Information Security, 2016

In many problems of combinatory analysis, operations of addition of sets are used (sum, direct sum, direct product etc.). In the present paper, as well as in the preceding one [1], some properties of addition operation of sets (namely, Minkowski addition) in Boolean space n B are presented. Also, sums and multisums of various "classical figures" as: sphere, layer, interval etc. are considered. The obtained results make possible to describe multisums by such characteristics of summands as: the sphere radius, weight of layer, dimension of interval etc. using the methods presented in [2], as well as possible solutions of the equation X Y A + =, where n X Y A B ⊆ , , , are considered. In spite of simplicity of the statement of the problem, complexity of its solutions is obvious at once, when the connection of solutions with constructions of equidistant codes or existence the Hadamard matrices is apparent. The present paper submits certain results (statements) which are to be the ground for next investigations dealing with Minkowski summation operations of sets in Boolean space.

Philosophical transactions. Series A, Mathematical, physical, and engineering sciences, 2006

This is a paper in mathematics, specifically in set theory. On the example of the measure recognition problem (MRP), the paper highlights the phenomenon of the utility of a multidisciplinary mathematical approach to a single mathematical problem, in particular, the value of a set-theoretic analysis. MRP asks if for a given Boolean algebra, B, and a property, Phi, of measures, one can recognize by purely combinatorial means if B supports a strictly positive measure with property Phi. The most famous instance of this problem is MRP (countable additivity), and in the first part of the paper, we survey the known results on this and some other problems. We show how these results naturally lead to asking about two other specific instances of the problem MRP, namely MRP (non-atomic) and MRP (separable). Then, we show how our recent work gives an easy solution to the former of these problems and some partial information about the latter. The long-term goal of this line of research is to obt...

2011

In this paper we introduce a novel, simpler form of the polytope of inner Bayesian approximations of a belief function, or \consistent probabilities". We prove that the set of vertices of this polytope is generated by all possible permutations of elements of the domain, mirroring a similar behavior of outer consonant approximations. An intriguing connection with the behavior of maximal outer consonant approximations is highlighted, and the notion of inner (outer) approximation of a credal set in terms of lower probabilities proposed. Finally, we generalize the main result to the case of k-additive belief functions, belief functions whose focal elements have size at most k. We prove that the set of such objects dominating a given belief function is also a polytope whose vertices are generated by permutations of focal elements of size at most k.

Pacific Journal of Mathematics, 1964

The results of this work constitute part of the author's dissertation done under the supervision of Prof. Tarski. The author wishes to thank Prof. Horn for his help in checking the proof of the main result.

Mathematische Zeitschrift, 2000

We investigate the vertex-connectivity of the graph of f -monotone paths on a d-polytope P with respect to a generic functional f . The third author has conjectured that this graph is always (d − 1)-connected. We resolve this conjecture positively for simple polytopes and show that the graph is 2-connected for any d-polytope with d ≥ 3. However, we disprove the conjecture in general by exhibiting counterexamples for each d ≥ 4 in which the graph has a vertex of degree two.

Journal of Mathematical Analysis and Applications, 1997

A set function is a function whose domain is the power set of a set, which is assumed to be finite in this paper. We treat a possibly nonadditive set function, i.e., Ž. Ž. a set function which does not satisfy necessarily additivity, A q B s Ž. AjB for A l B s л, as an element of the linear space on the power set. Then some of the famous classes of set functions are polyhedral in that linear space, i.e., expressed by a finite number of linear inequalities. We specify the sets of the coefficients of the linear inequalities for some classes of set Ž. functions. Then we consider the following three problems: a the domain Ž. extension problem for nonadditive set functions, b the sandwich problem for Ž. nonadditive set functions, and c the representation problem of a binary relation by a nonadditive set function, i.e., the problem of nonadditive comparative probabilities.

One classical result of Freimann gives the optimal lower bound for the cardinality of A + A if A is a d-dimensional finite set in R d . Matolcsi and Ruzsa have recently generalized this lower bound to |A + kB| if B is d-dimensional, and A is contained in the convex hull of B. We characterize the equality case of the Matolcsi-Ruzsa bound. The argument is based partially on understanding triangulations of polytopes.