Beitr\"age zur Algebra und Geometrie
Contributions to Algebra and Geometry
Volume 36 (1995), No. 2

H. Groemer:
On a Characterization of Circles and Spheres

Abstract
It is shown that, among all two-dimensional convex domains, circles
can be characterized by the property that for all circumscribed rectangles
the midpoints of the sides meet the domain. The proof uses Fourier series,
and this method yields not only such characterizations but also
corresponding stability results. Moreover, some analogous results for the
$d$-dimensional case $(d\ge 3)$ are established.