An i-hedrite is a 4-regular plane graph with faces of size 2, 3 and 4. We do a short survey of th... more 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
A non-Platonic convex polyhedron in R 3 is semiregular if it is vertex-transitive and its faces a... more 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 well-known GEM’s (graph-encoded maps) M on the sphere S 2. We collect in the Table 1 data on two relatively new embeddings of those maps: 1. the Euler characteristic χ of the surface, on which their maps skew(M) and phial(M) embed; 2. the hypercube embedding, if any, of the path-metric of the graph G(M). In this Note, an hypercube embedding of a map M means that 2d (doubled pathmetric of its skeleton G(M)) isometrically embeds into the path-metric of some halfcube 1 2 Hm. Isometric embedding of d in the path-metric of some hypercube Hm implies isometric embedding of 2d into a Johnson graph J(2m, m), which implies isometric embedding of 2d in ...
26.62> L 1 -metric 8.1 The L 1 -metric in probability theory 8.2 The ` 1 -metric in statistica... more 26.62> L 1 -metric 8.1 The L 1 -metric in probability theory 8.2 The ` 1 -metric in statistical data analysis 8.3 The ` 1 -metric in computer vision and pattern recognition 2 M. Deza and M. Laurent 1 Preliminaries We recall in this Section all the definitions that we need for this Chapter and, in particular, the definitions about distance spaces, isometric embeddings, measure spaces, and our main host spaces, namely, the Banach ` p - and L p -spaces for 1 p 1. 1.0.1 Distance spaces and ` p -spaces Let X be a set. A function d : X Theta X ! R+ is called a distance on X if d is symmetric, i.e.,
We address various topologies (de Bruijn, chordal ring, generalized Petersen, meshes) in various ... more 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 regular maps.
We describe in this chapter the other main known classes of valid inequalities defining facets of... more We describe in this chapter the other main known classes of valid inequalities defining facets of the cut polytope. The complete linear description of the cut polytope CUT n □ is known only for n £</font > 7n \le 7 ; it is presented in Section 30.6.
In this chapter we present several operations on valid inequalities and facets of the cut polytop... more In this chapter we present several operations on valid inequalities and facets of the cut polytope. One of the basic properties of the cut polytope CUT n □ is that all its facets can be deduced from the facets of the cut cone CUT n using the so-called switching operation (cf. Section 26.3.2). In fact, switchings and permutations constitute the
ABSTRACT We consider in this chapter several additional questions related to the notion of hyperc... more ABSTRACT We consider in this chapter several additional questions related to the notion of hypercube embedding. A possible way of relaxing this notion is to look for integer combinations rather than nonnegative integer combinations of cut semimetrics. In other words, one considers the lattice Ln\cal L_n generated by all cut semimetrics on V n . We recall in Section 25.1 the characterization of Ln\cal L_n . This is an easy result; namely, Ln\cal L_n consists of the integer distances satisfying the parity condition. We also present the characterization of some sublattices of Ln\cal L_n , namely, of the sublattice generated by all even T-cut semimetrics and of the sublattice generated by all κ-uniform cut semimetrics.
An i-hedrite is a 4-regular plane graph with faces of size 2, 3 and 4. We do a short survey of th... more 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
A non-Platonic convex polyhedron in R 3 is semiregular if it is vertex-transitive and its faces a... more 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 well-known GEM’s (graph-encoded maps) M on the sphere S 2. We collect in the Table 1 data on two relatively new embeddings of those maps: 1. the Euler characteristic χ of the surface, on which their maps skew(M) and phial(M) embed; 2. the hypercube embedding, if any, of the path-metric of the graph G(M). In this Note, an hypercube embedding of a map M means that 2d (doubled pathmetric of its skeleton G(M)) isometrically embeds into the path-metric of some halfcube 1 2 Hm. Isometric embedding of d in the path-metric of some hypercube Hm implies isometric embedding of 2d into a Johnson graph J(2m, m), which implies isometric embedding of 2d in ...
26.62> L 1 -metric 8.1 The L 1 -metric in probability theory 8.2 The ` 1 -metric in statistica... more 26.62> L 1 -metric 8.1 The L 1 -metric in probability theory 8.2 The ` 1 -metric in statistical data analysis 8.3 The ` 1 -metric in computer vision and pattern recognition 2 M. Deza and M. Laurent 1 Preliminaries We recall in this Section all the definitions that we need for this Chapter and, in particular, the definitions about distance spaces, isometric embeddings, measure spaces, and our main host spaces, namely, the Banach ` p - and L p -spaces for 1 p 1. 1.0.1 Distance spaces and ` p -spaces Let X be a set. A function d : X Theta X ! R+ is called a distance on X if d is symmetric, i.e.,
We address various topologies (de Bruijn, chordal ring, generalized Petersen, meshes) in various ... more 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 regular maps.
We describe in this chapter the other main known classes of valid inequalities defining facets of... more We describe in this chapter the other main known classes of valid inequalities defining facets of the cut polytope. The complete linear description of the cut polytope CUT n □ is known only for n £</font > 7n \le 7 ; it is presented in Section 30.6.
In this chapter we present several operations on valid inequalities and facets of the cut polytop... more In this chapter we present several operations on valid inequalities and facets of the cut polytope. One of the basic properties of the cut polytope CUT n □ is that all its facets can be deduced from the facets of the cut cone CUT n using the so-called switching operation (cf. Section 26.3.2). In fact, switchings and permutations constitute the
ABSTRACT We consider in this chapter several additional questions related to the notion of hyperc... more ABSTRACT We consider in this chapter several additional questions related to the notion of hypercube embedding. A possible way of relaxing this notion is to look for integer combinations rather than nonnegative integer combinations of cut semimetrics. In other words, one considers the lattice Ln\cal L_n generated by all cut semimetrics on V n . We recall in Section 25.1 the characterization of Ln\cal L_n . This is an easy result; namely, Ln\cal L_n consists of the integer distances satisfying the parity condition. We also present the characterization of some sublattices of Ln\cal L_n , namely, of the sublattice generated by all even T-cut semimetrics and of the sublattice generated by all κ-uniform cut semimetrics.
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