Rate of Approximation of Regular Convex Bodies by Convex Algebraic Level Surfaces

Abstract

We consider the problem of the approximation of regular convex bodies in ℝ^d by level surfaces of convex algebraic polynomials. Hammer (in Mathematika 10, 67–71, 1963) verified that any convex body in ℝ^d can be approximated by a level surface of a convex algebraic polynomial. In Jaen J. Approx. 1, 97–109, 2009 and subsequently in J. Approx. Theory 162, 628–637, 2010 a quantitative version of Hammer’s approximation theorem was given by showing that the order of approximation of convex bodies by convex algebraic level surfaces of degree n is $\frac{1}{n}$. Moreover, it was also shown that whenever the convex body is not regular (that is, there exists a point on its boundary at which the convex body possesses two distinct supporting hyperplanes), then $\frac{1}{n}$ is essentially the sharp rate of approximation. This leads to the natural question whether this rate of approximation can be improved further when the convex body is regular. In this paper we shall give an affirmative answer to this question. It turns out that for regular convex bodies a o(1/n) rate of convergence holds. In addition, if the body satisfies the condition of C ²-smoothness the rate of approximation is $O(\frac{1}{n^{2}})$.