Orlicz Centroid Bodies

The sharp affine isoperimetric inequality that bounds the volume of the centroid body of a star body (from below) by the volume of the star body itself is the Busemann-Petty centroid inequality. A decade ago, the Lp analogue of the classical BusemannPetty centroid inequality was proved. Here, the definition of the centroid body is extended to an Orlicz centroid body of a star body, and the corresponding analogue of the Busemann-Petty centroid inequality is established for convex bodies. The centroids of the intersections of an origin-symmetric body with half-spaces form the surface of a convex body. This “centroid body” is a concept that dates back at least to Dupin. The classical affine isoperimetric inequality that relates the volume of a convex body with that of its centroid body was conjectured by Blaschke and established in a landmark work by Petty [53]. Petty’s inequality became known as the Busemann-Petty centroid inequality because, in establishing his inequality, Petty not only made critical use of Busemann’s random simplex inequality, but as Petty stated, he “reinterpreted” it. (See, e.g., the books by Gardner [12], Leichtweiss [25], Schneider [55], and Thompson [58] for reference.) The concept of a centroid body had a natural extension in what became known as the Lp Brunn-Minkowski theory and its dual. This theory had its origins in the early 1960s when Firey (see, e.g., Schneider [55]) introduced his concept of Lp compositions of convex bodies. Three decades later, in [34] and [35] these Firey-Minkowski Lp combinations were shown to lead to an embryonic Lp Brunn-Minkowski theory. This theory (and its dual) has witnessed a rapid growth. (See, e.g., [1–9,17– 23,26–32,34–44,46–48,54,56,57,59,62].) The Lp analogues of centroid bodies became a central focus within the Lp Brunn-Minkowski theory and its dual and establishing the Lpanalogue of the Busemann-Petty centroid inequality became a major goal. This was accomplished by the authors of the present paper in [37] with an independent approach presented by Campi and Gronchi [3]. The Lp centroid bodies quickly became objects of interest in asymptotic Mathematics Subject Classification. MR 52A40. Research supported, in part, by NSF Grant DMS–0706859. PROOF COPY 1 NOT FOR DISTRIBUTION 2 ERWIN LUTWAK, DEANE YANG AND GAOYONG ZHANG geometric analysis (see, e.g., [10], [11], [24], [49], [50], [51], [52]) and even the theory of stable distributions (see, e.g., [48]). Using concepts introduced by Ludwig [29], Haberl and Schuster [21] were led to establish “asymmetric” versions of the Lp Busemann-Petty centroid inequality that, for bodies that are not origin-symmetric, are stronger than the Lp Busemann-Petty centroid inequality presented in [37] and [3]. The “asymmetric” inequalities obtained by Haberl and Schuster are ideally suited for non-symmetric bodies. This can be seen by looking at the Haberl-Schuster version of the Lp analogue of the classical Blaschke-Santaló inequality that was presented in [46]. While for origin symmetric bodies, the Lp extension of [46] does recover the original Blaschke-Santaló inequality as p→∞, for arbitrary bodies only the Haberl-Schuster version does so. The works of Haberl and Schuster [21] (see also [22]), Ludwig and Reitzner [32], and Ludwig [31] have demonstrated the clear need to move beyond the Lp Brunn-Minkowski theory to what we are calling an Orlicz Brunn-Minkowski theory. This need is not only motivated by compelling geometric considerations (such as those presented in Ludwig and Reitzner [32]), but also by the desire to obtain Sobolev bounds (see [22]) of a far more general nature. This paper is the second in a series intended to develop a few of the elements of an Orlicz Brunn-Minkowski theory and its dual. Here we define the Orlicz centroid body, establish some of its basic properties, and most importantly establish (what we call) the Orlicz BusemannPetty centroid inequality (for Orlicz centroid bodies). We consider convex φ : R → [0,∞) such that φ(0) = 0. This means that φ must be decreasing on (−∞, 0] and increasing on [0,∞). We require that one of these is happening strictly so; i.e., φ is either strictly decreasing on (−∞, 0] or strictly increasing on [0,∞). The class of such φ will be denoted by C. If K is a star body (see Section 1 for precise definitions) with respect to the origin in Rn with volume |K|, and φ ∈ C then we define the Orlicz centroid body ΓφK of K as the convex body whose support function at x ∈ Rn is given by h(ΓφK;x) = inf { λ > 0 : 1 |K| ∫ K φ (x · y λ ) dy ≤ 1 } , where x ·y denotes the standard inner product of x and y in Rn and the integration is with respect to Lebesgue measure in Rn. When φp(t) = |t|p, with p ≥ 1, then