BRUNN–MINKOWSKI TYPE INEQUALITIES FOR L_p MOMENT BODIES

Abstract

About 15 years ago, Lutwak and Zhang (E. Lutwak and G. Zhang, Blaschke-Santalo inequalities, J. Differ. Geom. 47 (1997), 1-16) introduced the notion of L p moment bodies and established important volume inequalities for them, which were recently generalized by Haberl and Schuster (C. Haberl and E. Schuster, General L p affine isoperimetric inequalities, J. Differ. Geom. 83 (2009), 1-26). In this paper, we establish new Brunn-Minkowski-type inequalities with respect to Blaschke L p harmonic addition for the quermassintegrals and dual quermassintegrals of L p moment bodies.2000 Mathematics Subject Classification. 52A40. Introduction and main results.Centroid bodies were defined and investigated by Petty [27]. They have proven to be a remarkably powerful tool in establishing a number of fundamental affine isoperimetric inequalities due to Petty [27][28][29] (see also [18,19]). Projection bodies were introduced by Minkowski at the turn of the previous century and have since become a central notion in convex geometry (see e.g.[8] and the references therein). All centroid and projection bodies are zonoids. However, the centroid operator and the projection operator are quite different. For example, the centroid operator commutes with linear transformations, while the projection operator does not; the projection operator is translation invariant, but the centroid operator is not.While projection bodies and the projection operator have been the objects of intensive investigations during more than 50 years, centroid bodies (volumenormalized moment bodies) and their L p extensions by Lutwak and Zhang [24] have attracted increased attention only in the last two decades (see e.g. [4-6, 8-15, 19-25, 33, 35, 36, 38]). The aim of this paper is to obtain the Brunn-Minkowski-type inequalities with respect to Blaschke L p harmonic combinations for quermassintegrals and dual quermassintegrals of L p moment bodies (see e.g. [1, 2, 26, 31, 32] for related results).Let K n denote the set of convex bodies (compact, convex subsets with non-empty interiors) in ‫ޒ‬ n . For the set of convex bodies containing the origin in their interior and the set of convex bodies centred at the origin, we write K n o and K n s respectively. The unit ball in ‫ޒ‬ n and its surface will be denoted by B and S n−1 respectively. The volume and surface area of the convex body K will be denoted by V (K) and S(K) respectively.