On the approximation of convex bodies by convex algebraic level surfaces

In this note we consider the problem of the approximation of convex bodies in RdRd by level surfaces of convex algebraic polynomials. Hammer (1963) [1] verified that any convex body in RdRd can be approximated by a level surface of a convex algebraic polynomial. In Kroó (2009) [3] 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 nn is bounded from above by clognn. In this paper we improve further this approximation result by verifying an upper bound of order 1n. Moreover, it will be also shown that this bound is sharp, in general.