Convex Geometry

WHY A MATHEMATICS OF UNCERTAINTY?

- probabilities do not represent well ignorance and lack of data;
- evidence is normally limited, rather than infinite as assumed by (frequentist) probability;
- expert knowledge needs often to be combined with hard evidence;
- in extreme cases (rare events or far-future predictions) very little data;
- bottom line: not enough evidence to determine the actual probability describing the problem.

The principal aim of this book is to introduce to the widest possible audience an original view of belief calculus and uncertainty theory. In this geometric approach to uncertainty, uncertainty measures can be seen as points of a suitably complex geometric space, and manipulated in that space, for example, combined or conditioned.

In the chapters in Part I, Theories of Uncertainty, the author offers an extensive recapitulation of the state of the art in the mathematics of uncertainty. This part of the book contains the most comprehensive summary to date of the whole of belief theory, with Chap. 4 outlining for the first time, and in a logical order, all the steps of the reasoning chain associated with modelling uncertainty using belief functions, in an attempt to provide a self-contained manual for the working scientist. In addition, the book proposes in Chap. 5 what is possibly the most detailed compendium available of all theories of uncertainty. Part II, The Geometry of Uncertainty, is the core of this book, as it introduces the author’s own geometric approach to uncertainty theory, starting with the geometry of belief functions: Chap. 7 studies the geometry of the space of belief functions, or belief space, both in terms of a simplex and in terms of its recursive bundle structure; Chap. 8 extends the analysis to Dempster’s rule of combination, introducing the notion of a conditional subspace and outlining a simple geometric construction for Dempster’s sum; Chap. 9 delves into the combinatorial properties of plausibility and commonality functions, as equivalent representations of the evidence carried by a belief function; then Chap. 10 starts extending the applicability of the geometric approach to other uncertainty measures, focusing in particular on possibility measures (consonant belief functions) and the related notion of a consistent belief function. The chapters in Part III, Geometric Interplays, are concerned with the interplay of uncertainty measures of different kinds, and the geometry of their relationship, with a particular focus on the approximation problem. Part IV, Geometric Reasoning, examines the application of the geometric approach to the various elements of the reasoning chain illustrated in Chap. 4, in particular conditioning and decision making. Part V concludes the book by outlining a future, complete statistical theory of random sets, future extensions of the geometric approach, and identifying high-impact applications to climate change, machine learning and artificial intelligence.

The book is suitable for researchers in artificial intelligence, statistics, and applied science engaged with theories of uncertainty. The book is supported with the most comprehensive bibliography on belief and uncertainty theory.

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, 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.

The principal aim of this book is to introduce to the widest possible audience an original view of belief calculus and uncertainty theory. In this geometric approach to uncertainty, uncertainty measures can be seen as points of a suitably complex geometric space, and manipulated in that space, for example, combined or conditioned.

In the chapters in Part I, Theories of Uncertainty, the author offers an extensive recapitulation of the state of the art in the mathematics of uncertainty. This part of the book contains the most comprehensive summary to date of the whole of belief theory, with Chap. 4 outlining for the first time, and in a logical order, all the steps of the reasoning chain associated with modelling uncertainty using belief functions, in an attempt to provide a self-contained manual for the working scientist. In addition, the book proposes in Chap. 5 what is possibly the most detailed compendium available of all theories of uncertainty. Part II, The Geometry of Uncertainty, is the core of this book, as it introduces the author’s own geometric approach to uncertainty theory, starting with the geometry of belief functions: Chap. 7 studies the geometry of the space of belief functions, or belief space, both in terms of a simplex and in terms of its recursive bundle structure; Chap. 8 extends the analysis to Dempster’s rule of combination, introducing the notion of a conditional subspace and outlining a simple geometric construction for Dempster’s sum; Chap. 9 delves into the combinatorial properties of plausibility and commonality functions, as equivalent representations of the evidence carried by a belief function; then Chap. 10 starts extending the applicability of the geometric approach to other uncertainty measures, focusing in particular on possibility measures (consonant belief functions) and the related notion of a consistent belief function. The chapters in Part III, Geometric Interplays, are concerned with the interplay of uncertainty measures of different kinds, and the geometry of their relationship, with a particular focus on the approximation problem. Part IV, Geometric Reasoning, examines the application of the geometric approach to the various elements of the reasoning chain illustrated in Chap. 4, in particular conditioning and decision making. Part V concludes the book by outlining a future, complete statistical theory of random sets, future extensions of the geometric approach, and identifying high-impact applications to climate change, machine learning and artificial intelligence.

The book is suitable for researchers in artificial intelligence, statistics, and applied science engaged with theories of uncertainty. The book is supported with the most comprehensive bibliography on belief and uncertainty theory.

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 $C$ to its convex hull $K$. We show that if $C$ contains the relative boundary of $K$, and $f$ is a continuous convex function on $C$, then $f$ extends to a continuous convex function on $K$ using the standard convex roof construction. In fact, a necessary and sufficient condition for $f$ to extend from any set to a continuous convex function on the convex hull is that $f$ 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 $C$ does not contain the entire relative boundary of $K$, then there may not exist any continuous convex extension of $f$. Finally, when $\partial K$ and $f$ are $C^1$ we give a necessary and sufficient condition for the convex roof construction to be $C^1$ on all of $K$. 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, 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 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, 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 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.

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

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 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.