Caracte´risation des ellipsoı¨des par leurs groupes d'automorphismes

This article investigates automorphisms of strongly convex bodies of <f>Rn</f> that are convex bodies with a boundary of class <f>C2</f> whose Hessian is non-degenerate everywhere. A real version of B. Wong's [11] theorem is obtained: the automorphism group of a strongly convex body is compact unless its boundary is an ellipsoid. This result is also in line with the result [8] of J. Lelong-Ferrand. Our second theorem completes a rigidity result of Benze´cri [3]: the interior of a strongly convex body is the universal covering of a non-simply connected locally projective manifold if and only if it is the interior of an ellipsoid. [Copyright &y& Elsevier]