On the approximation of convex bodies by convex algebraic level surfaces
\textit{H. Minkowski} [Math. Ann. 57, 447--495 (1903; JFM 34.0649.01)] showed that the boundary of any convex body in \(\mathbb{R}^d\), can be approximated arbitrarily well by a level surface of a convex analytic function. \textit{P. C. Hammer} [Mathematika 10, 64--71 (1963; Zbl 0122.41004)] generalized Minkowski's result by showing that his approximation result remains valid for approximation by convex algebraic level surfaces. In [\textit{A. Kroó}, Jaen J. Approx. 1, No.~1, 97--109 (2009; Zbl 1183.41010)] Hammer's approximation theorem was quantified by showing that the order of approximation of convex surfaces by convex algebraic level surfaces of degree \(n\) is bounded from above by \({c\log n\over n}\) and it was also conjectured that the \(\log n\) term in this upper bound can be omitted. In this paper, the author shows that the boundary of every convex body in \(\mathbb{R}^d\) can be approximated by level surfaces of convex algebraic polynomials of degree at most \(n\) with order \(1/n\). It is also shown that in a certain sense this order of approximations is the best possible.
scientific article; zbMATH DE number 5706953