1. Basic convexity 2. Boundary structure 3. Minkowski addition 4. Curvature measure and quermass integrals 5. Mixed volumes 6. Inequalities for mixed volumes 7. Selected applications Appendix.

/pdf/convex-bodies-the-brunn-minkowski-theory-3raynvod45.pdf

Convex bodies : the Brunn-Minkowski theory

Quantization is a process that maps a continous or discrete set of values into approximations that belong to a smaller set. Quantization is a lossy: some information about the original data is lost in the process. The key to a successful quantization is therefore the selection of an error criterion – such as entropy and signal-to-noise ratio – and the development of optimal quantizers for this criterion.

Quantization

We describe a program to prove the Kepler conjecture on sphere packings. We then carry out the first step of this program. Each packing determines a decomposition of space into Delaunay simplices, which are grouped together into finite configurations called Delaunay stars. A score, which is related to the density of packings, is assigned to each Delaunay star. We conjecture that the score of every Delaunay star is at most the score of the stars in the face-centered cubic and hexagonal close packings. This conjecture implies the Kepler conjecture. To complete the first step of the program, we show that every Delaunay star that satisfies a certain regularity condition satisfies the conjecture.

Sphere packings I

Quantization, the process of approximating continuous-amplitude signals by digital (discrete-amplitude) signals, is an important aspect of data compression or coding, the field concerned with the reduction of the number of bits necessary to transmit or store analog data, subject to a distortion or fidelity criterion. The independent quantization of each signal value or parameter is termed scalar quantization, while the joint quantization of a block of parameters is termed block or vector quantization. This tutorial review presents the basic concepts employed in vector quantization and gives a realistic assessment of its benefits and costs when compared to scalar quantization. Vector quantization is presented as a process of redundancy removal that makes effective use of four interrelated properties of vector parameters: linear dependency (correlation), nonlinear dependency, shape of the probability density function (pdf), and vector dimensionality itself. In contrast, scalar quantization can utilize effectively only linear dependency and pdf shape. The basic concepts are illustrated by means of simple examples and the theoretical limits of vector quantizer performance are reviewed, based on results from rate-distortion theory. Practical issues relating to quantizer design, implementation, and performance in actual applications are explored. While many of the methods presented are quite general and can be used for the coding of arbitrary signals, this paper focuses primarily on the coding of speech signals and parameters.

Vector quantization in speech coding

https://annals.math.princeton.edu/wp-content/uploads/annals-v162-n3-p01.pdf

A proof of the Kepler conjecture

Lagerungen in der Ebene auf der Kugel und im Raum

where d denotes the diameter of the circumcircle of the points. The constant cannot be replaced by a greater one. In order to prove our theorem we denote the points by P~ . . . . ,P~,Fixing the points ~ . . . . , P,,, we consider S,, as a function of the point P~. Since the distances P~P., , . . . , P1P,~ are convex functions of P~, the same can b e 'stated of S,i. It follows that S,~ takes its maximum for a point P1 ly!ng on the boundary C of the circumcircle of the points. Therefore all points may be supposed to lie on C. Furthermore we suppose that C is a unit circle and that the cyclical order of the points is PI . . . . . P,,,. Introducing the notations P,+I = P1, . . . , P~,~ 1 P,,-~ we consider the sum

On the sum of distances determined by a pointset

Let J be a system of sets. If all members of J are contained in a given set C and each point of C belongs to at most one member of J then J is said to be a packing into C. If, on the other hand, each point of C belongs to at least one member of J then we say that J is a covering of C. Packings and coverings have been considered in various spaces and on various combinatorial structures. Here we are interested in problems concerning packings and coverings consisting of convex bodies in spaces of constant curvature, i.e. in Euclidean, spherical and hyperbolic space. Instead of saying that J is a packing into the whole space or J is a covering of the whole space we shall simply use the terms J is a packing and J is a covering.

New Results in the Theory of Packing and Covering

http://students.mimuw.edu.pl/~tt249057/other/2306047.pdf

L. Fejes Tóth

Papers

Lagerungen in der Ebene auf der Kugel und im Raum

On the sum of distances determined by a pointset

New Results in the Theory of Packing and Covering

On the Densest Packing of Spherical Caps

Sur la représentation d'une population infinie par un nombre fini d'éléments