Geometriae Dedicata 63: 267-296, 1996. 267 (E) 1996 Kluwer Academic Publishers. Printed in the Netherlands. The Cross-Section Body, Plane Sections of Convex Bodies and Approximation of Convex Bodies, I E. MAKAI, JR. t * and H. MARTINI 2 I Mathematical Institute of the Hungarian Academy of Sciences, Pf. 127, H-1364 Budapest, Hungary. e-mail: makai@math-inst.hu 2Technische Universit& Chemnitz-Zwickau, Fakult~it fiir Mathematik, Postfach 964, D-09009 Chemnitz, Germany. e-mail: martini@mathematik.tu-chemnitz.de (Received: 25 February 1994; revised version: 27 December 1995) Abstract. For a convex body K C ]~d we investigate three associated bodies, its intersection body IK (for 0 C int K), cross-section body CK, and projection body HK, which satisfy IK C CK C IIK. Conversely we prove CK C consh(d)I(K - x) for some x E intK, and HK C const2 (d)CK, for certain constants, the first constant being sharp. We estimate the maximal k-volume of sections of ½(K + (-K)) with k-planes parallel to a fixed k-plane by the analogous quantity for K; our inequality is, if only k is fixed, sharp. For L C ~a a convex body, we take n random segments in L, and consider their 'Minkowski average' D. We prove that, for V(L) fixed, the supremum of V(D) (with also n E N arbitrary) is minimal for L an ellipsoid. This result implies the Petty projection inequality about max V((IIM)*), for M C ]~a a convex body, with V(M) fixed. We compare the volumes of projections of convex bodies and the volumes of the projections of their sections, and, dually, the volumes of sections of convex bodies and the volumes of sections of their circumscribed cylinders. For fixed n, thepth moments of V(D) (1 < p < c~) also are minimized, for V(L) fixed, by the ellipsoids. For k = 2, the supremum (n E N arbitrary) and the pth moment (n fixed) of V(D) are maximized for example by triangles, and, for L centrally symmetric, for example by parallelograms. Last we discuss some examples for cross-section bodies. Mathematics Subject Classifications (1991): 52A20, 52A22, 52A27, 52A40. Key words: projection body, intersection body, cross-section body, central symmetral, ellipsoid, zonoid, Petty projection inequality. 1. Basic Notions Let K C ~d be a convex body, i.e. a compact, convex set with non-empty interior. We always suppose d > 2. In this paper we will investigate several bodies associated with K. We denote by Vk the k-dimensional Lebesgue measure, for k = d we write V instead of Ira. For K C ~d a convex body and x E intK the radial function rx of K with respect to x is the strictly positive function on S a-t for which bd K = { x + rx ( u )u I u E S a-l}. Centred means: symmetric about the origin. For x, g E I~ d the segment with endpoints x, y is denoted by [x, y]. * Research (partially) supported by Hungarian National Foundation for Scientific Research, Grant No. 41.