Convex Geometry Research Papers - Academia.edu (original) (raw)

Let V be a flnite set and M a flnite collection of subsets of V. Then M is an alignment of V if and only if M is closed under taking intersections and contains both V and the empty set. If M is an alignment of V , then the elements of M... more

Let V be a flnite set and M a flnite collection of subsets of V. Then M is an alignment of V if and only if M is closed under taking intersections and contains both V and the empty set. If M is an alignment of V , then the elements of M are called convex sets and the pair (V;M) is called an aligned space or a convexity space. If S µ V , then the convex hull of S, denoted by CH(S), is the smallest convex set that contains S. Suppose X 2 M. Then x 2 X is an extreme point for X if X ¡ fxg 2 M. A convex geometry on a flnite set is an aligned space with the additional property that every convex set is the convex hull of its extreme points. This property is referred to as the Minkowski-Krein-Milman property. The distance between a pair of vertices u;v of G is the number of edges in a u ¡ v geodesic (i.e., a shortest u ¡ v path) in G and is denoted by dG(u;v) or d(u;v). The interval between a pair u;v of vertices in a graph G, denoted by IG(u;v) or I(u;v), is the collection of all vertices...

A function F defined on all subsets of a finite ground set E is quasi-concave if F(X∪ Y)≥{F(X),F(Y)} for all X,Y⊂ E. Quasi-concave functions arise in many fields of mathematics and computer science such as social choice, theory of graph,... more

A function F defined on all subsets of a finite ground set E is quasi-concave if F(X∪ Y)≥{F(X),F(Y)} for all X,Y⊂ E. Quasi-concave functions arise in many fields of mathematics and computer science such as social choice, theory of graph, data mining, clustering and other fields. The maximization of quasi-concave function takes, in general, exponential time. However, if a quasi-concave function is defined by associated monotone linkage function then it can be optimized by the greedy type algorithm in a polynomial time. Quasi-concave functions defined as minimum values of monotone linkage functions were considered on antimatroids, where the correspondence between quasi-concave and bottleneck functions was shown (Kempner & Levit, 2003). The goal of this paper is to analyze quasi-concave functions on different families of sets and to investigate their relationships with monotone linkage functions.

We study piecewise linear approximation of quadratic functions de- flned on Rn. Invariance properties and canonical Caley/Klein metrics that help in understanding this problem can be handled in arbitrary dimensions. However, the problem... more

We study piecewise linear approximation of quadratic functions de- flned on Rn. Invariance properties and canonical Caley/Klein metrics that help in understanding this problem can be handled in arbitrary dimensions. However, the problem of optimal approximants in the sense that their linear pieces are of maximal size by keeping a given error tolerance, is a di-cult one. We present a

We study continuity and regularity of convex extensions of functions from a compact set CCC to its convex hull KKK. We show that if CCC contains the relative boundary of KKK, and fff is a continuous convex function on CCC, then fff... more

We study continuity and regularity of convex extensions of functions from a compact set CCC to its convex hull KKK. We show that if CCC contains the relative boundary of KKK, and fff is a continuous convex function on CCC, then fff extends to a continuous convex function on KKK using the standard convex roof construction. In fact, a necessary and sufficient condition for fff to extend from any set to a continuous convex function on the convex hull is that fff extends to a continuous convex function on the relative boundary of the convex hull. We give examples showing that the hypotheses in the results are necessary. In particular, if CCC does not contain the entire relative boundary of KKK, then there may not exist any continuous convex extension of fff. Finally, when partialK\partial KpartialK and fff are C1C^1C1 we give a necessary and sufficient condition for the convex roof construction to be C1C^1C1 on all of KKK. We also discuss an application of the convex roof construction in quantum computation.

In this paper we extend our geometric approach to the theory of evidence in order to include other important classes of finite fuzzy measures. In particular we describe the geometric counterparts of possibility measures or fuzzy sets,... more

In this paper we extend our geometric approach to the theory of evidence in order to include other important classes of finite fuzzy measures. In particular we describe the geometric counterparts of possibility measures or fuzzy sets, represented as consonant belief functions. The correspondence between chains of subsets and convex sets of consonant functions is studied and its properties analyzed, eventually yielding an elegant representation of the region of consonant belief functions in terms of the notion of simplicial complex.

In this paper we will prove that if a compact AAA in RnR^nRn belongs to the unit ball in RnR^nRn, then AAA has a slice of measure greater than a calculable constant times the measure of AAA. Our result is sharp.

In his work on log-concavity of multiplicities, Okounkov showed in passing that one could associate a convex body to a linear series on a projective variety, and then use convex geometry to study such linear systems. Although Okounkov was... more

In his work on log-concavity of multiplicities, Okounkov showed in passing that one could associate a convex body to a linear series on a projective variety, and then use convex geometry to study such linear systems. Although Okounkov was essentially working in the classical setting of ample line bundles, it turns out that the construction goes through for an arbitrary big divisor. Moreover, this viewpoint renders transparent many basic facts about asymptotic invariants of linear series, and opens the door to a number of extensions. The purpose of this paper is to initiate a systematic development of the theory, and to give a number of applications and examples.

In this paper we extend our geometric approach to the theory of evidence in order to include other important classes of nite fuzzy measures. In particular we describe the geometric counterparts of possibility measures or fuzzy sets,... more

In this paper we extend our geometric approach to the theory of evidence
in order to include other important classes of nite fuzzy measures.
In particular we describe the geometric counterparts of possibility measures or fuzzy sets, represented as consonant belief functions.
The correspondence between chains of subsets and convex sets of consonant functions is studied and its properties analyzed, eventually yielding an elegant representation of the region of consonant belief functions
in terms of the notion of simplicial complex.

In this work we extend the geometric approach to the theory of evidence in order to study the geometric behavior of the two quantities inherently associated with a belief function. i.e. the plausibility and commonality functions. After... more

In this work we extend the geometric approach to the theory of evidence in order to study the geometric behavior of the two quantities inherently associated with a belief function. i.e. the plausibility and commonality functions. After introducing the analogous of the basic probability assignment for plausibilities and commonalities, we exploit it to understand the simplicial form of both plausibility and commonality spaces. Given the intuition provided by the binary case we prove the congruence of belief, plausibility, and commonality spaces for both standard and unnormalized belief functions, and describe the point-wise geometry of these sum functions in terms of the rigid transformation mapping them onto each other. This leads us to conjecture that the D-S formalism may be in fact a geometric calculus in the line of geometric probability, and opens the way to a wider application of discrete mathematics to subjective probability.

Convexity in Graphs and Hypergraphs. [SIAM Journal on Algebraic and Discrete Methods 7, 433 (1986)]. Martin Farber, Robert E. Jamison. Abstract. We study several notions of abstract convexity in graphs and hypergraphs. In ...

The purpose of this paper is to investigate the interplay arising between max algebra, convexity and scaling problems. The latter, which have been studied in nonnegative matrix theory, are strongly related to max algebra. One problem is... more

The purpose of this paper is to investigate the interplay arising between max algebra, convexity and scaling problems. The latter, which have been studied in nonnegative matrix theory, are strongly related to max algebra. One problem is strict visualisation scaling, which means finding, for a given nonnegative matrix A, a diagonal matrix X such that all elements of X^{-1}AX are

This is an overview of merging the techniques of vector lattice theory and convex geometry.

We establish some new quantitative results on Steiner/Schwarz-type symmetrizations, continuing the line of results from [Bourgain et al. (Lecture Notes in Math. 1376 (1988), 44–66)] on Steiner symmetrizations. We show that if we... more

We establish some new quantitative results on Steiner/Schwarz-type symmetrizations, continuing the line of results from [Bourgain et al. (Lecture Notes in Math. 1376 (1988), 44–66)] on Steiner symmetrizations. We show that if we symmetrize high-dimensional sections of convex bodies, then very few steps are required to bring such a body close to a Euclidean ball.

We introduce and study a new class of -convex bodies (extending the class of convex bodies) in metric and normed linear spaces. We analyze relations between characteristic properties of convex bodies, demonstrate how -convex bodies... more

We introduce and study a new class of -convex bodies (extending the class of convex bodies) in metric and normed linear spaces. We analyze relations between characteristic properties of convex bodies, demonstrate how -convex bodies connect with some classical results of Convex Geometry, as Helly theorem, and find applications to geometric tomography. We introduce the notion of a circular projection and investigate the problem of determination of -convex bodies by their projection-type images. The results generalize corresponding stability theorems by H. Groemer.