Some \(\log\)-Minkowski inequalities for \(L_p\)-mixed affine surface area
The \(L_p\) Brunn-Minkowski theory is an important theory in convex geometry which has received widespread attention; the related results are very rich when \(p\geq1\). Since the outstanding works of Böröczky, Lutwak, Yang and Zhang [\textit{K. Böröczky jun.} et al., Adv. Math. 231, No. 3--4, 1974--1997 (2012; Zbl 1258.52005); J. Am. Math. Soc. 26, No. 3, 831--852 (2013; Zbl 1272.52012)], the \(L_p\) Brunn-Minkowski theory obtains increasing attention when \(p<1\). Two major open problems in this field are the so-called conjectured \(\log\)-Brunn-Minkowski inequality and the conjectured \(\log\)-Minkowski inequality. In the present paper, the authors define the \(L_p\)-mixed affine surface area probability measure and establish the \(\log\)-Minkowski inequalities for non-symmetric convex bodies of \(L_p\)-mixed affine surface area. The idea used in this paper is inspired by the work of \textit{A. Stancu} [Adv. Appl. Math. 73, 43--58 (2016; Zbl 1331.52014)].
\(L_p\)-mixed affine surface area probability measure
scientific article; zbMATH DE number 7382086
scientific article