Geometriae Dedicata 72: 63–68, 1998. 63 c 1998 Kluwer Academic Publishers. Printed in the Netherlands. Approximation of Convex Bodies by Centrally Symmetric Bodies MAREK LASSAK Instytut Matematyki i Fizyki ATR, 85-796 Bydgoszcz, Poland e-mail: lassak@atr.bydgoszcz.pl (Received: 8 July 1997) Abstract. We present an analog of the well-known theorem of F. John about the ellipsoid of maximal volume contained in a convex body. Let be a convex body and let be a centrally symmetric convex body in the Euclidean -space. We prove that if is an affine image of of maximal possible volume contained in , then a subset of the homothetic copy of with the ratio 2 1 and the homothety center in the center of . The ratio 2 1 cannot be lessened as a simple example shows. Mathematics Subject Classifications (1991): 52A20, 52A21. Key words: convex body, approximation, affine transformation, volume, Banach–Mazur distance. Denote by the Euclidean -space. If is a convex body, then by we mean the homothetic copy of of ratio and the center in the centroid of . In particular, if is centrally symmetric, then the centroid is the center of symmetry of . For every convex body there is exactly one ellipsoid (i.e., an affine image of the unit ball) of maximum volume contained in . The celebrated theorem of F. John [3] says that . Our aim is to give an analog of this theorem when the ball is replaced by an arbitrary centrally symmetric convex body. This analogy is even more natural when this centrally symmetric body is treated as the unit ball of a finite-dimensional real normed space. The case of a simplex in the part of shows that the ratio in John’s theorem cannot be lessened. The ratio 2 1 in the theorem presented below cannot be lessened as well; we take a simplex as and a parallelotope as . This is explained after the proof. Let us add that the theorem was announced in [5]. THEOREM. Let be a convex body and let be a centrally symmetric convex body. If is an affine image of with maximum possible volume contained in , then 2 1 . Proof. If 1, then the Theorem is obvious. Assume that 2. Denote by the smallest positive number such that 2 1 . In order to prove the required inclusion, we will show that . Suppose the opposite, i.e., suppose that .