Sharp inequalities related to the functional $$U_j$$ U j and some applications
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Sharp inequalities related to the functional Uj and some applications Ai-Jun Li1 · Si-Tao Zhang1 Received: 24 June 2019 / Accepted: 16 October 2020 © Springer Nature B.V. 2020
Abstract Sharp inequalities of the parameterized functional U j for Borel measures on the unit sphere in Rn are established. As two applications, some inequalities related to cone-volume measures and Schneider’s projection problem are obtained. Keywords The parameterized functional U j · Isotropic measure · Cone-volume measure · Schneider’s projection problem Mathematics Subject Classification (2000) 52A40
1 Introduction Let K be a convex body (compact convex set with non-empty interior) in Euclidean space Rn . The support function h K of K is defined by h K (x) = max{x · y : y ∈ K }, where x · y denotes the usual inner product of x and y in Rn . For x ∈ Rn , we denote the Euclidean norm of x by x. For each Borel set ω on the unit sphere S n−1 in Rn , the surface area measure S K (ω) of K is the (n − 1)-dimensional Hausdorff measure of the set of all boundary points of K for which there exists a normal vector of K belonging to ω. If K contains the origin in its interior, then the cone-volume measure, VK , of K is a Borel measure on S n−1 defined by VK (ω) =
ω
h K (u) d S K (u). n
(1.1)
The first author was supported by Key Research Project for Higher Education in Henan Province (No. 17A110022).
B
Ai-Jun Li [email protected] Si-Tao Zhang [email protected]
1
School of Mathematics and Information Science, Henan Polytechnic University, Jiaozuo City 454000, China
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Geometriae Dedicata
Obviously, the total mass of the cone-volume measure |VK | of K is the volume of K , denoted by V (K ). In recent years, cone-volume measures were intensively studied in various contexts, see, e.g., [5–11,16,17,19–21,25,26,31,42–48]. In particular, cone-volume measures are the subjects of the logarithmic Minkowski problem [10], which is the particular interesting case p = 0 of the general L p -Minkowski problem that is at the core of the L p -Brunn–Minkowski theory, began from L p Minkowski–Firey combinations [14] in the 1960’s and came to life when Lutwak [28,29] introduced the concept of L p surface area measure in the 1990’s. For the detailed bibliography on this topic we refer the reader to [39, Chapter 9]. A fundamental object in convex geometric analysis is the projection body K of a convex body K in Rn whose support function is given, for u ∈ S n−1 , by 1 h K (u) = |u · v|d S K (v). (1.2) 2 S n−1 An important unsolved problem regarding projection bodies is Schneider’s projection problem (see, e.g., [38–40]): what is the least upper bound, as K ranges over the class of origin-symmetric convex bodies in Rn , of the affine-invariant ratio [V (K )/V (K )n−1 ]1/n . Schneider [38] conjectured that this ratio is maximized by parallelotopes. However, a counterexample was presented in [13] to show that this is not the case. This problem is still open now. As an effective tool in studying Schneider’s projection problem, Lutwak et al. [31] introd
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