AbstractThis paper presents bounds for the best approximation, with respect to the Hausdorff metric, of a convex bodyKby a circumscribed polytopePwith a given number of facets. These bounds are of particular interest ifKis elongated. To measure the elongation of the convex set, its isoperimetric ratioVj(K)1/jVi(K)−1/iis used.
In this paper we use the concept of numerical range to characterize best
approximation points in closed convex subsets of B(H): Finally by using this
method we give also a useful characterization of best approximation in
closed convex subsets of a C*-algebra A.
AbstractWe introduce and investigate an adaptation of Fourier series to set-valued functions (multifunctions, SVFs) of bounded variation. In our approach we define an analogue of the partial sums of the Fourier series with the help of the Dirichlet kernel using the newly defined weighted metric integral. We derive error bounds for these approximants. As a consequence, we prove that the sequence of the partial sums converges pointwisely in the Hausdorff metric to the values of the approximated set-valued function at its points of continuity, or to a certain set described in terms of the metric selections of the approximated multifunction at a point of discontinuity. Our error bounds are obtained with the help of the new notions of one-sided local moduli and quasi-moduli of continuity which we discuss more generally for functions with values in metric spaces.
Polytopal approximation of elongated convex bodies
