The Hausdorff nearest circle to a convex compact set in the plane (Q1393109)

Let \(T\subset \mathbb{E}^2\) be a plane convex body (compact). The authors show that there exists precisely one plane disk \(K\) such that the Hausdorff distance between \(T\) and \(K\) is minimal by applying theorems on Chebyshev approximation to the support function of \(T\). The position of \(K\) is characterized by geometrical properties. In the case that \(T\) is a polygon the construction of \(K\) is considered in detail. The construction of \(K\) yields inequalities for the Hausdorff distance of \(T\) and \(K\) in terms of the diameter of \(T\) and a particular value of its width. Finally, the Remez algorithm to calculate the parameters of \(K\) numerically is cited and interpreted geometrically. Moreover, an algorithm is given which finds \(K\) in the case of a polygon \(T\) in finitely many steps.