CONVEX BODIES WITH HOMOTHETIC SECTIONS LUIS MONTEJANO ABSTRACT We prove that if K is a convex body in E n+1 , n ^ 2, and p 0 is a point of K with the property that all bi- sections of K through p 0 are homothetic, then K is a Euclidean ball. 1. Introduction Let AT be a convex body in E" +1 , n ^ 2, and let p 0 be a point of K. Suppose that all «-sections of K through p 0 are affinely equivalent. Must K be an ellipsoid? If n is even, Gromov [4] and independently Mani [5] and Burton [2] proved that the answer is yes. The problem is still unsolved when n is odd. Suppose now that all «-sections of K through p 0 are congruent. If n is even, Gromov's result implies that K is an ellipsoid and hence it is easy to check that K is a Euclidean ball; if n = 3, Burton [2] proved that K is a Euclidean ball; and finally, Schneider [7] gave a proof of the same statement for n ^ 2. The purpose of this paper is to prove the following results. THEOREM 1. If all n-sections of K through p 0 are affinely equivalent, then either K is an ellipsoid or K is centrally symmetric with respect to p 0 . THEOREM 2. If all n-sections of K through p 0 are volume-preserving affinely equivalent, then K is a Euclidean ball. As a corollary, we have Schneider's Theorem. COROLLARY. If all n-sections ofK through p 0 are congruent, then K is a Euclidean ball. THEOREM 3. If all n-sections ofK through p 0 are homothetic, then K is a Euclidean ball. The corresponding results about ^-sections, 2 ^ k ^ n, follow immediately from the above results or their proofs. 2. Definitions and preliminaries Let 7 n+1 be the group of isometries of E n+1 , the (n+l)-dimensional Euclidean space, and let O n+1 <= 7 n+1 be the orthogonal subgroup. We choose an orthonormal base JC 1} x 2 ,..., x n+1 of E n+1 ; denote by E* the subspace spanned by its first k vectors, Received 18 June 1990; revised 28 November 1990. 1980 Mathematics Subject Classification 52A20, 55R10. Research supported by the Alexander von Humboldt Foundation. Bull. London Math. Soc. 23 (1991) 381-386