A Helmholtz-Lie type characterization of ellipsoids. II
The authors prove the Theorem: A closed convex surface \(S\) in Euclidean \(d\)-space \(E^d\) (:= boundary of a compact convex subset of \(E^d\) with nonempty interior) is an ellipsoid if and only if for any \(x,y\in S\) there is an affinity \(A_{xy}\) mapping \(x\) onto \(y\) and a neighbourhood of \(x\) in \(S\) onto a neighbourhood of \(y\) in \(S\). Using topological tools this theorem was already proved for odd \(d\) by the first author in the earlier part I of this paper [ibid. 13, No. 3-4, 517-527 (1995; Zbl 0824.52006)]. Here the proof for all \(d\) \((\geq 2)\) is given. It relies strongly on results on floating bodies (Leichtweiss) and in affine differential geometry (Blaschke, Petty). A characterization of ellipsoids due to Petty is needed utilizing the notions of affine distance and Santaló point. In the case \(d=2\) the solution of a functional equation is needed. As an important step in the case \(d\geq 3\), the authors now could prove (which was to be assumed additionally in a first version of this article) that then \(|\text{det }A_{xy}|=1\) for all \(x,y\in S\).
scientific article; zbMATH DE number 919856