The Hausdorff Nearest Circle to a Triangle

Abstract

The problem of finding the nearest in the Hausdorif metrics circle to a non-empty convex compact set $T$ in the plane is considered. It is shown that this problem is equivalent to the problem of the Chebyshevian best approximation of $2\pi$-periodic functions by trigonometric polynomials of first order and in consequence the Hausdorif nearest circle is unique. The case when $T$ is a triangle is solved completely. It is shown then that the center of the nearest circle is the intersection point of the midline of the longest side and the bisectrix against the shortest side of the triangle.