A general isepiphanic inequality

An inequality of Petty regarding the volume of a convex body and that of the polar of its projection body is shown to lead to an inequality between the volume of a convex body and the power means of its brightness function. A special case of this power-mean inequality is the classical isepiphanic (isoperimetric) inequality. The power-mean inequality can also be used to obtain strengthened forms and extensions of some known and conjectured geometric inequalities. Affine projection measures (Quermassintegrale) are introduced. In [12] it was shown that the Blaschke-Santalo inequality [23] leads immediately to a power-mean inequality relating the volume of a convex body and the power means of its width (function). Special cases of this power-mean inequality include the classical inequalities of Urysohn and Bieberbach. It will be shown in the present note that an inequality of Petty [19], which we will refer to as the Petty projection inequality, leads immediately to an analogous power-mean inequality relating the volume of a convex body and the power means of its brightness (function). A special case of this power-mean inequality is the classical isepiphanic (isoperimetric) inequality. This power-mean inequality also leads to inequalities similar to some width-volume inequalities obtained by Chakerian [6, 7], Chakerian and Sangwine-Yager [8], and the author [15]. When combined with otf r known inequalities, this power-mean inequality can be used to obtain a strengthened form of an inequality of Knothe [11] and Chakerian [5] relating the volume of a convex body and the arithmetic mean of the volumes of its circumscribed right cylinders. Finally, it solves completely a problem posed in [14], and can be used to prove two (similar) conjectures of the author [16, 26]. The setting for this note is Euclidean n-dimensional space, R' (n > 2). We will use the letter K (possibly with subscripts) to denote a convex body (compact convex set with nonempty interior) in RI. We use S` to denote the surface and w, to denote the n-dimensional volume of the unit ball in R'. The letter u will denote a unit vector, exclusively. For a given direction u E Sn1, we use Eu to denote the hyperplane (passing through the origin) orthogonal to u. For a given K and u E S"n, we use bK(u) and cK(U) to denote respectively the width and brightness of K in the direction u; i.e., LK(U) is the (n 1)-dimensional volume of the projection of K onto Eu, while bK(U) is the 1-dimensional volume of the projection of K onto the orthogonal complement of Eu. For the volume, surface area, and mean width of K, we write V(K), S(K), and B(K), respectively. The reader is referred to [3 and 9] for material relating to convex bodies. Received by the editors January 12, 1983. 1980 Mathematics Subject Classification. Primary 52A40; Secondary 53A15, 46B20.