Analytic continuation of convex bodies and Funk's characterization of the sphere. (Q700620)

This paper contains the following theorem related to P. Funk's 1916 characterization of the sphere [Math. Ann. 77, 129--135 (1915; JFM 45.0533.01)]. Let \(K\) be an origin-centred star body in \({\mathbb R}^n\) and let \(\mathcal H\) be an open connected set of hyperplanes through the origin covering \({\mathbb R}^n\). If \(K\) has constant cross-sectional areas when sliced by hyperplanes from \(\mathcal H\), and for some hyperplane \(H\in{\mathcal H}\), \(K\) osculates a ball centred at the origin to infinite order along \(H\), then \(K\) coincides with the ball. Without the osculation hypothesis there are counterexamples. It is shown that for each \(k\in{\mathbb N}\) and \(0<\theta<\pi/2\) there exists a smooth and strictly convex \(K\) that is not a ball but is origin-centred, osculates a ball to order \(k\) along the hyperplane \(x_n=0\), and has constant cross-sectional areas when sliced by any hyperplane making an angle of at most \(\theta\) with the \(x_1\)-axis. Similar results are shown for projection areas. The proofs use the Radon and cosine transforms.