In this paper, we focus on lower bounds and algorithms for some basic geometric problems in the one-pass (insertion only) streaming model. The problems considered are grouped into three categories — (i) Klee’s measure (ii) Convex body approximation, geometric query, and (iii) Discrepancy Klee’s measure is the problem of finding the area of the union of hyperrectangles. Under convex body approxi...

For some applications it is desired to approximate a set of m data points in IR with a convex quadratic function. Furthermore, it is required that the convex quadratic approximation underestimate all m of the data points. It is shown here how to formulate and solve this problem using a convex quadratic function with s = (n+ 1)(n+ 2)/2 parameters, s ≤ m, so as to minimize the approximation error...

This paper deals with sparse approximations by means of convex combinations of elements from a predetermined \basis" subset S of a function space. Speciically, the focus is on the rate at which the lowest achievable error can be reduced as larger subsets of S are allowed when constructing an approximant. The new results extend those given for Hilbert spaces by Jones and Barron, including in par...

Estimating the degree of approximation in the uniform norm, of a convex function on a finite interval, by convex algebraic polynomials, has received wide attention over the last twenty years. However, while much progress has been made especially in recent years by, among others, the authors of this article, separately and jointly, there have been left some interesting open questions. In this pa...