On a characterization of circles and spheres (Q1903358)

If a rectangle is circumscribed about a circular disc, the disc will always touch each side of the rectangle at its midpoint. The author shows that this property characterizes the circular disc among all two-dimensional convex domains. Moreover, if the point of contact of a convex domain with the edge of every circumscribing rectangle is always close to the midpoint (or if the quadratic mean distance is small), then the domain is close to a disc in the Hausdorff metric (resp. quadratic mean metric). This is also generalized to higher dimensions; a convex body \(K\) in \(E^d\) is a ball if and only if it touches the center of every facet of every circumscribed box. Again, if the centers of the facets of circumscribed boxes are close to the points of contact with \(K\), then \(K\) is close to a ball in the Hausdorff metric.