Some questions of the theory of approximations of functions and sets in the Hausdorff metric

B. Kh. Sendov

Trudy Matematicheskogo Instituta imeni V.A. Steklova, Tome 24 (1969) no. 5, pp. 143-183

Résumé

The paper is a survey of the literature on the theory of approximation in the Hausdorff metric, and certain related questions. In the first chapter the definition of the Hausdorff distance is given, together with some of its properties. The relation between the Hausdorff and uniform distances is also discussed. The second chapter gives a survey of results relating to the calculation of $\varepsilon$-entropy, $\varepsilon$-capacity, and widths relative to the Hausdorff distance. A central position is occupied by the third chapter, where a number of estimates are given of the best approximation of functions and curves in the plane relative to the Hausdorff distance. A theorem is proved here on the existence of a universal estimate of the best approximation relative to the Hausdorff distance for all bounded functions. The question of the approximation of convex functions and curves by polygons, relative to the uniform and Hausdorff distances, is treated separately. Chapter 4 is devoted to linear approximations relative to the Hausdorff distance and the convergence of sequences of positive and convex linear operators. In a short final chapter a new problem in the theory of approximations is proposed.