Michel Marie Deza

Ecole Normale Supérieure de Paris | ENS · Laboratoire Interdisciplinaire de Géométrie Applique

The Niemeier lattices are the 23 unimodular even lattices of norm 2 in dimension 24. We determine computationally their covering radius for 16 of those lattices and give lower bounds for the remaining that we conjecture to be exact. This is achieved by computing the list of Delaunay polytopes of those lattices.

Zigzags and generalized zigzags in thin chamber complexes are investigated, in particular, all zigzags in the Coxeter complexes are described. Using this description, we show that the lengths of all generalized zigzags in the simplex $\alpha_{n}$, the cross-polytope $\beta_{n}$, the $24$-cell, the icosahedron and the $600$-cell are equal to the Cox...

Given a connected surface with Euler characteristic and three integers , an - is a -embedded graph, having vertices of degree only k and only a- and b-gonal faces. The main case are (geometric) fullerenes (5, 6; 3)-. By , we denote the number of a-gonal, b-gonal faces. Call an -map lego-admissible if either , or is integer. Call it lego-like if it...

A convex body in the n-dimensional Euclidean space \(\mathbb{E}^{n}\) is a convex compact connected subset of \(\mathbb{E}^{n}\). It is called solid (or proper) if it has nonempty interior. Let K denote the space of all convex bodies in \(\mathbb{E}^{n}\), and let K p be the subspace of all proper convex bodies. Given a set \(X \subset \mathbb{E}^{...

Image Processing treats signals such as photographs, video, or tomographic output. In particular, Computer Graphics consists of image synthesis from some abstract models, while Computer Vision extracts some abstract information: say, the 3D description of a scene from video footage of it. From about 2000, analog image processing (by optical devices...

Distances are mainly used in Biology to pursue basic classification tasks, for instance, for reconstructing the evolutionary history of organisms in the form of phylogenetic trees. In the classical approach those distances were based on comparative morphology, physiology, mating studies, paleaontology and immunodiffusion. The progress of modern Mol...

A group (G, ⋅ , e) is a set G of elements with a binary operation ⋅ , called the group operation, that together satisfy the four fundamental properties of closure (x ⋅ y ∈ G for any x, y ∈ G), associativity (x ⋅ (y ⋅ z) = (x ⋅ y) ⋅ z for any x, y, z ∈ G), the identity property (\(x \cdot e = e \cdot x = x\) for any x ∈ G), and the inverse property...

An alphabet is a finite set \(\mathcal{A}\), \(\vert \mathcal{A}\vert \geq 2\), elements of which are called characters (or symbols). A string (or word) is a sequence of characters over a given finite alphabet \(\mathcal{A}\). The set of all finite strings over the alphabet \(\mathcal{A}\) is denoted by \(W(\mathcal{A})\). Examples of real world ap...

A probability space is a measurable space \((\varOmega,\mathcal{A},P)\), where \(\mathcal{A}\) is the set of all measurable subsets of Ω, and P is a measure on \(\mathcal{A}\) with P(Ω) = 1. The set Ω is called a sample space. An element \(a \in \mathcal{A}\) is called an event. P(a) is called the probability of the event a. The measure P on \(\mat...

Physics studies the behavior and properties of matter in a wide variety of contexts, ranging from the submicroscopic particles from which all ordinary matter is made (Particle Physics) to the behavior of the material Universe as a whole (Cosmology).

Functional Analysis is the branch of Mathematics concerned with the study of spaces of functions. This usage of the word functional goes back to the calculus of variations which studies functions whose argument is a function. In the modern view, Functional Analysis is seen as the study of complete normed vector spaces, i.e., Banach spaces.

In this chapter we consider a special class of metrics defined on some normed structures, as the norm of the difference between two given elements. This structure can be a group (with a group norm), a vector space (with a vector norm or, simply, a norm), a vector lattice (with a Riesz norm), a field (with a valuation), etc.

Some immediate generalizations of the notion of metric, for example, quasi-metric, near-metric, extended metric, were defined in Chap. 1 Here we give some generalizations in the direction of Topology, Probability, Algebra, etc.

Riemannian Geometry is a multidimensional generalization of the intrinsic geometry of 2D surfaces in the Euclidean space \(\mathbb{E}^{3}\). It studies real smooth manifolds equipped with Riemannian metrics, i.e., collections of positive-definite symmetric bilinear forms ((g ij )) on their tangent spaces which vary smoothly from point to point. The...

The Universe is defined as the whole space-time continuum in which we exist, together with all the energy and matter within it.

Coding Theory deals with the design and properties of error-correcting codes for the reliable transmission of information across noisy channels in transmission lines and storage devices. The aim of Coding Theory is to find codes which transmit and decode fast, contain many valid code words, and can correct, or at least detect, many errors. These ai...

Any L p-metric (as well as any norm metric for a given norm | | . | | on \(\mathbb{R}^{2}\)) can be used on the plane \(\mathbb{R}^{2}\), and the most natural is the L 2-metric, i.e., the Euclidean metric \(d_{E}(x,y) = \sqrt{(x_{1 } - y_{1 } )^{2 } + (x_{2 } - y_{2 } )^{2}}\) which gives the length of the straight line segment [x, y], and is the i...

A network is a graph, directed or undirected, with a positive number (weight) assigned to each of its arcs or edges. Real-world complex networks usually have a gigantic number N of vertices and are sparse, i.e., with relatively few edges.

A data set is a finite set comprising m sequences (x 1 j , …, x n j ), j ∈ { 1, …, m}, of length n. The values x i 1, …, x i m represent an attribute S i .

Given a finite set A of objects A i in a space S, computing the Voronoi diagram of A means partitioning the space S into Voronoi regions V (A i ) in such a way that V (A i ) contains all points of S that are “closer” to A i than to any other object A j in A.

The term length has many meanings: distance, extent, linear measure, span, reach, end, limit, etc.; for example, the length of a train, a meeting, a book, a trip, a shirt, a vowel, a proof. The length of an object is its linear extent, while the height is the vertical extent, and width (or breadth) is the side-to-side distance at 90∘ to the length,...

A topological space (X, τ) is a set X with a topology τ, i.e., a collection of subsets of X with the following properties: 1. X ∈ τ, ∅ ∈ τ; 2. If A, B ∈ τ, then A ∩ B ∈ τ; 3. For any collection {A α }α , if all A α ∈ τ, then ∪α A α ∈ τ.

A distance space (X, d) is a set X (carrier) equipped with a distance d.

Geometry arose as the field of knowledge dealing with spatial relationships. It was one of the two fields of pre-modern Mathematics, the other being the study of numbers.

There are many ways to obtain new distances (metrics) from given distances (metrics). Metric transforms give new distances as a functions of given metrics (or given distances) on the same set X. A metric so obtained is called a transform metric. We give some important examples of transform metrics in Sect. 4.1.

A graph is a pair G = (V, E), where V is a set, called the set of vertices of the graph G, and E is a set of unordered pairs of vertices, called the edges of the graph G. A directed graph (or digraph) is a pair D = (V, E), where V is a set, called the set of vertices of the digraph D, and E is a set of ordered pairs of vertices, called arcs of the...

In this chapter we group the main distances used in Systems Theory (such as Transition Systems, Dynamical Systems, Cellular Automata, Feedback Systems) and other interdisciplinary branches of Mathematics, Engineering and Theoretical Computer Science (such as, say, Robot Motion and Multi-objective Optimization).

A surface is a real 2D (two-dimensional) manifold M 2, i.e., a Hausdorff space, each point of which has a neighborhood which is homeomorphic to a plane \(\mathbb{E}^{2}\), or a closed half-plane (cf. Chap. 7).

In Geography, spatial scales are shorthand terms for distances, sizes and areas. For example, micro, meso, macro, mega may refer to local (0.001–1), regional (1–100), continental (100–10,000), global ( > 10,000) km, respectively.

Here we consider the most important metrics on the classical number systems: the semiring \(\mathbb{N}\) of natural numbers, the ring \(\mathbb{Z}\) of integers, and the fields \(\mathbb{Q}\), \(\mathbb{R}\), \(\mathbb{C}\) of rational, real, complex numbers, respectively. We consider also the algebra \(\mathcal{Q}\) of quaternions.

In this chapter we present selected distances used in real-world applications of Human Sciences. In this and the next chapter, the expression of distances ranges from numeric (say, in m) to ordinal (as a degree assigned according to some rule) and nominal.

In this chapter we group together distances and distance paradigms which do not fit in the previous chapters, being either too practical (as in equipment), or too general, or simply hard to classify.

We consider zigzags in thin complexes. The main result states that the sum of the lengths of all zigzags in an $n$-complexe is equal to the sum of the lengths of all zigzags in all $(n-1)$-faces of this complex, and this sum also is the twice of the sum of the lengths of all zigzags in all $(n-2)$-faces. For simplicial and cubical $n$-complexes, th...

A binary linear code is called {\em LCD} if it intersects its dual trivially. We show that the coefficients of the joint weight enumerator of such a code with its dual satisfy linear constraints, leading to a new linear programming bound on the size of an LCD code of given length and minimum distance. In addition, we show that this polynomial is, i...

We address various topologies (de Bruijn, chordal ring, generalized Petersen, meshes) in various ways ( isometric embedding, embedding up to scale, embedding up to a distance) in a hypercube or a half-hypercube. Example of obtained embeddings: infinite series of hypercube embeddable Bubble Sort and Double Chordal Rings topologies, as well as of reg...

In this chapter, we consider parametrization and, especially, one with \(1\) complex parameter, i.e., the Goldberg–Coxeter construction \(GC_{k,l}(G_0)\) (a generalization of a simplicial subdivision of Dodecahedron considered in [Gold37] and [Cox71]), producing a plane graph from any \(3\)- or \(4\)-regular plane graph \(G_0\) for integer paramete...

We consider the zigzag and railroad structures of \(3\)-regular plane graphs and, especially, graphs \(a_v\), i.e., \(v-vertex\) \((\{a,6\},3)\)-spheres, where \(a=2\), \(3\), or \(4\). The case \(a=5\) has been treated in previous Chapter.

The fullerenes , i.e., the maps \((\{5,6\},3)\)-\(\mathbb {S}^2\), are of particular interest in Carbon Chemistry. Denote by \(F_v(G)\) any \(v\)-vertex fullerene of symmetry \(G\). Denote by \(C_v\) and call IP fullerene any \(F_v\) with isolated (i.e., no two of them are adjacent) \(5\)-gons. A number of \(C_v\)’s with \(60\le v<100\), including...

For a \(4\)-regular plane graph without loops, Euler formula \(v-e+f=2\) takes the form.

In this chapter, based mainly on [DDS13a] and [DeDu12], we consider \((\{3,4\}, 5)\)-spheres (named icosahedrites) and \((\{1, 2,3\}, 6)\)-spheres. Both cases allow to consider zigzags. But in contrast to the \(3\)- and \(4\)-regular cases, the second structure of edges appear: weak zigzags for \(5\)- and central circuits for \(6\)-regular graphs.

In this chapter, based mainly on [DeDu04], we focus on generalization of zigzags for higher dimension. Inspired by Coxeter’s notion of Petrie polygon for \(d\)-polytopes (see [Cox73]), we generalize the notion of zigzag circuits on complexes and compute the zigzag structure for several interesting families of \(d\)-polytopes, including semiregular,...

In this chapter, \(ZC\)-(zigzag or central circuit) structure will be mainly described using groups. But first, Propositions 7.1 and 7.2 below treat the easiest case: \(ZC\)-structure of the \(k\)-inflation \(GC_{k,0}(G_0)\) of \(G_0\) in terms of \(ZC\)-structure of \(G_0\); see example in Fig. 7.5.

In this chapter we summarize the main notions considered in this book and, briefly, the results that we obtain. Specifically, we define the pure graph theoretic and plane graph theoretic notions needed for this work with emphasis on symmetries.

The lists of facets -- $298,592$ in $86$ orbits -- and of extreme rays -- $242,695,427$ in $9,003$ orbits -- of the hypermetric cone $HYP_8$ are computed. The first generalization considered is the hypermetric polytope $HYPP_n$ for which we give general algorithms and a description for $n\le 8$. Then we shortly consider generalizations to simplices...

We report here a computation giving the complete list of facets for the cut polytopes over several very symmetric graphs with $15-30$ edges, including $K_8$, $K_{3,3,3}$ $K_{1,4,4}$, $K_{5,5}$, some other $K_{l,m}$, $K_{1,l,m}$, $Prism_7, APrism_6$, M\"{o}bius ladder $M_{12}$, Icosahedron, Heawood and Petersen graphs. For $K_8$, it shows that the h...

Geometry arose as the field of knowledge dealing with spatial relationships. It was one of the two fields of pre-modern Mathematics, the other being the study of numbers.

A polyhedral norm is a norm N on R^n for which the set N(x)\leq 1 is a polytope. This covers the case of the L^1 and L^{\infty} norms. We consider here effective algorithms for determining the Voronoi polytope for such norms with a point set being a lattice. The algorithms, that we propose, use the symmetries effectively in order to compute a decom...

The walk distances in graphs have no direct interpretation in terms of walk weights, since they are introduced via the \emph{logarithms} of walk weights. Only in the limiting cases where the logarithms vanish such representations follow straightforwardly. The interpretation proposed in this paper rests on the identity $\ln\det B=\tr\ln B$ applied t...

A geometric fullerene, or simply a fullerene, is the surface of a simple closed convex 3-dimensional polyhedron with only 5- and 6-gonal faces. Fullerenes are geometric models for chemical fullerenes, which form an important class of organic molecules. These molecules have been studied intensively in chemistry, physics, crystallography, and so on,...

A d-code in a graph is a set of vertices such that all pairwise distances are at least d. As part of a study of d-codes of three-and four-dimensional regular polytopes, the maximum independent set order of the 120-cell is calculated. A linear program based on counting arguments leads to an upper bound of 221. An independent set of order 110 in the...

A geometric fullerene, or simply a fullerene, is the surface of a simple closed convex 3-dimensional polyhedron with only 5-and 6-gonal faces. Fullerenes are geometric models for chemical fullerenes, which form an important class of organic molecules. These molecules have been studied intensively in chemistry, physics, crystallography, and so on, a...

This chapter discusses the concepts related to distances in Coding Theory. Coding Theory deals with the design and the properties of error-correcting codes for the reliable transmission of information across noisy channels in transmission lines and storage devices. The aim of Coding Theory is to find codes that transmit and decode fast, contain man...

Distances are mainly used in Biology to pursue basic classification tasks, for instance, for reconstructing the evolutionary history of organisms in the form of phylogenetic trees. In the classical approach those distances were based on comparative morphology, physiology, mating studies, paleontology and immunodiffusion. The progress of modern Mole...

A network is a graph, directed or undirected, with a positive number (weight) assigned to each of its arcs or edges. Real-world complex networks usually have a gigantic number N of vertices and are sparse, i.e., with relatively few edges.

Image Processing treats signals such as photographs, video, or tomographic output. In particular, Computer Graphics consists of image synthesis from some abstract models, while Computer Vision extracts some abstract information: say, the 3D (i.e., 3-dimensional) description of a scene from video footage of it. From about 2000, analog image processi...

In this chapter we present selected distances used in real-world applications of Human Sciences. In this and the next chapter, the expression of distances ranges from numeric (say, in m) to ordinal (as a degree assigned according to some rule). Depending on the context, the distances are either practical ones, used in daily life and work outside of...

Riemannian Geometry is a multidimensional generalization of the intrinsic geometry of two-dimensional surfaces in the Euclidean space \(\mathbb{E}^{3}\). It studies real smooth manifolds equipped with Riemannian metrics, i.e., collections of positive-definite symmetric bilinear forms ((g ij )) on their tangent spaces which vary smoothly from point...

Given a finite set A of objects A i in a space S, computing the Voronoi diagram of A means partitioning the space S into Voronoi regions V (A i ) in such a way that V (A i ) contains all points of S that are “closer” to A i than to any other object A j in A.

An alphabet is a finite set \(\mathcal{A}\), \(|\mathcal{A}| \ge 2\), elements of which are called characters (or symbols). A string (or word) is a sequence of characters over a given finite alphabet \(\mathcal{A}\). The set of all finite strings over the alphabet \(\mathcal{A}\) is denoted by \(W(\mathcal{A})\). Examples of real world applications...

This chapter provides an overview of how the concept of distance is used in physics and chemistry. Physical forces that act at a distance (that is, a push or pull which acts without physical contact) are nuclear and molecular attraction, and, beyond atomic level, gravity, static electricity, and magnetic forces. Last two forces can be both, push an...

A group (G, ⋅ , e) is a set G of elements with a binary operation ⋅ , called the group operation, that together satisfy the four fundamental properties of closure (x ⋅ y ∈ G for any x, y ∈ G), associativity (x ⋅ (y ⋅ z) = (x ⋅ y) ⋅ z for any x, y, z ∈ G), the identity property (\(x \cdot e = e \cdot x = x\) for any x ∈ G), and the inverse property...

Any L p -metric (as well as any norm metric for a given norm | | . | | on \(\mathbb{R}^{2}\)) can be used on the plane \(\mathbb{R}^{2}\), and the most natural is the L 2-metric, i.e., the Euclidean metric \(d_{E}(x,y) = \sqrt{(x_{1 } - y_{1 } )^{2 } + (x_{2 } - y_{2 } )^{2}}\) which gives the length of the straight line segment [x, y], and is the...

In this chapter we group together distances and distance paradigms which do not fit in the previous chapters, being either too practical (as in equipment), or too general, or simply hard to classify.

This chapter focuses on generalizations of metric spaces. If X be a set, then a function d : X3 → ℝ is called 2-metric if d is non-negative, totally symmetric, zero conditioned, and satisfies the tetrahedron inequality d (xl, x2, x3)≤ d (x4, x2, x3) + d (x1, x4, x3) + d (xl, x2, x4). It is the most important case m = 2 of the m-hemi-metric. A regul...

This chapter focuses on concept of distances in functional analysis. Functional Analysis is the branch of mathematics, concerned with the study of spaces of functions. This usage of the word functional goes back to the calculus of variations, implying a function whose argument is a function. In the modern view, functional analysis is seen as the st...

This chapter explains concepts related to distances in geometry. In mathematics, the notion of “geodesic” is a generalization of the notion of “straight line” to curved spaces. This term is taken from Geodesy, the science of measuring the size and the shape of the Earth. A geodesic is a curve that extends indefinitely in both directions and behaves...

A probability space is a measurable space\((\varOmega, \mathcal{A}, P)\), where \(\mathcal{A}\) is the set of all measurable subsets of Ω, and P is a measure on \(\mathcal{A}\) with P(Ω)=1. The set Ω is called a sample space. An element \(a \in \mathcal{A}\) is called an event. In particular, an elementary event is a subset of Ω that contains only...

This chapter provides an overview of units of length measurement and scales. The main length measure systems are Metric, Imperial (British and American), Japanese, Thai, Chinese Imperial, and Typographical. The International Metric System (or SI, short for Systeme International) is a modernized version of the metric system of units, established by...

Given R\subset N, an (R,k)$-sphere is a k-regular map on the sphere whose faces have gonalities i\in R. The most interesting/useful are (geometric) fullerenes, i.e., (\{5,6\},3)$-spheres. Call \kappa_i=1 + \frac{i}{k} - \frac{i}{2} the curvature of i-gonal faces. (R,k)-spheres admitting \kappa_i<0 are much harder to study. We consider the symmetrie...

The famous Deuxième mémoire of Voronoi (1908, 1909) in Crelle Journal contains, between other things, deep study of two dual partitions of R n related to an n-dimensional lattice Λ. In modern terms, they are called Voronoi partition and Delone partition (Voronoi himself called the second one L-partition). Both partitions coincide for the cubic latt...

A Frank–Kasper structure is a 3-periodic tiling of the Euclidean space E 3 by tetrahedra such that the vertex figure of any vertex belongs to four specified fullerenes with, respectively, 12, 14, 15, and 16 faces. Frank–Kasper structures occur in the crystallography of metallic alloys, clathrates, zeolites, and in geometrical optimization. 27 such...

We consider here 6-regular plane graphs whose faces have size 1, 2 or 3. In Section 2 a practical enumeration method is given that allowed us to enumerate them up to 53 vertices. Subsequently, in Section 3 we enumerate all possible symmetry groups of the spheres that showed up. In Section 4 we introduce a new Goldberg-Coxeter construction that take...

We consider 6-regular plane graphs whose faces have size 1,2 or 3. In Section 2 a practical enumeration method is given that allowed us to enumerate them up to 53 vertices. Subsequently, in Section 3 we enumerate all possible symmetry groups of the spheres that showed up. In Section 4 we introduce a new Goldberg-Coxeter construction that takes a 6-...

We consider 6-regular plane graphs whose faces have size 1, 2 or 3. In Section 2 a practical enumeration method is given that allowed us to enumerate them up to 53 vertices. Subsequently, in Section 3 we enumerate all possible symmetry groups of the spheres that showed up. In Section 4 we introduce a new Goldberg-Coxeter construction that takes a 6...

We show that the cone of weighted n-point quasi-metrics WQMet_n, the cone of weighted quasi-hypermetrics WHyp_n and the cone of oriented cuts OCut_n are projec- tions along an extreme ray of the metric cone Metn+1, of the hypermetric cone Hypn+1 and of the cut cone Cut_{n+1}, respectively. This projection is such that if one knows all faces of an o...

A partial semimetric on V_n={1, ..., n} is a function f=((f_{ij})): V_n^2 -> R_>=0 satisfying f_ij=f_ji >= f_ii and f_ij+f_ik-f_jk-f_ii >= 0 for all i,j,k in V_n. The function f is a weak partial semimetric if f_ij >= f_ii is dropped, and it is a strong partial semimetric if f_ij >= f_ii is complemented by f_ij <= f_ii+f_jj. We describe the cones o...

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We introduce the concept of protometric and present some properties of protometrics.

A non-Platonic convex polyhedron in R 3 is semiregular if it is vertex-transitive and its faces are regular polygons. The semiregular polyhedra consist of the 13 Archimedean solids and two infinite families of n-sided prisms Prismn and antiprisms APrismn. The skeleton graphs (vertices-edges) G(M) of the semiregular polyhedra and their duals are wel...

Frank-Kasper structures occur in the metallurgy of alloys, the crystallography of clathrates, in soap froths and in the solution to the weak Kelvin conjecture. By using a new combinatorial enumeration algorithm, 71 new structures have been found.

A Frank-Kasper structure is a 3-periodic tiling of the Euclidean space E3 by tetrahedra such that the vertex figure of any vertex belongs to four specified patterns with, respectively, 20, 24, 26 and 28 faces. Frank-Kasper structures occur in the crystallography of metallic alloys and clathrates. A new computer enumeration method has been devised f...

Finding the number 2H6 of directed Hamiltonian cycles in 6-cube is problem 43 in Section 7.2.1.1 of Knuth's ' The Art of Computer Programming'; various proposed estimates are surveyed below. We computed exact value: H6=14,754,666,508,334,433,250,560=6*2^4*217,199*1,085,989*5,429,923. Also the number Aut6 of those cycles up to automorphisms of 6-cub...

In combinatorics, the concept of Euclidean t-design was first defined by Neumaier-Seidel (1988) [25], as a two-step generalization of the concept of spherical t-design. It is possible to regard Euclidean t-design as a special case of general cubature ...

We group in this chapter several results concerning hypermetricity of distance spaces arising from graphs.

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An i-hedrite is a 4-regular plane graph with faces of size 2, 3 and 4. We do a short survey of their known properties (Deza et al. Proceedings of ICM Satellite Conference On Algebra and Combinatorics, 2003b; Deza et al. J Math Res Expo 22:49,2002; Deza and Shtogrin, Polyhedra in Science and Art 11:27, 2003a) and explain some new algorithms that all...

IntroductionKernel Elementary PolycyclesClassification of Elementary ({2, 3, 4, 5}, 3)-PolycyclesClassification of Elementary ({2, 3}, 4)-PolycyclesClassification of Elementary ({2, 3}, 5)-PolycyclesConclusion Appendix 1: 204 Sporadic Elementary ({2,3,4,5},3)-PolycyclesAppendix 2: 57 Sporadic eLementary ({2, 3}, 5)-polycyclesReferences

A connected graph is said to be ℓ1 if its path distance isometrically embeds into the space ℓ1. Following the work of Deza, Grishukhin, Shtogrin, and others on polyhedral ℓ1 graphs, we determine all finite closed polyhexes (trivalent surface graphs with hexagonal faces) that are ℓ1.

We find the allowed point-group symmetries of the cubic polyhedra with face sizes restricted to 3, 4, 5 and 6. For each group and face signature (p 3 , p 4 , p 5), a polyhedron with the smallest possible number of vertices is identified.

Distance metrics and distances have become an essential tool in many areas of pure and applied Mathematics, and this encyclopedia is the first one to treat the subject in full. The book appears just as research intensifies into metric spaces and especially, distance design for applications. These distances are particularly crucial, for example, in...