A new approach to the Minkowski first mixed volume and the LYZ conjecture
For a convex body \(K\) in \(\mathbb{R}^n\) having the origin in its interior, \textit{K. Böröczky jun.} et al. [J. Differ. Geom. 99, No. 3, 407--442 (2015; Zbl 1325.28004)] defined the cone-volume functional as \[ U(K)=\frac{1}{n^n}\int_{u_1 \wedge \cdots\wedge u_n\neq0}h_K(u_1)\cdots h_K(u_n)dS_K(u_1)\cdots dS_K(u_1), \] where \(h_K\) and \(S_K\) denote the support function and surface area measure of \(K\), respectively. Indeed, the cone-volume functional \(U\) was first defined by \textit{E. Lutwak} et al. [Trans. Am. Math. Soc. 353, No. 5, 1767--1779 (2001; Zbl 0971.52011)] for convex polytope in \(\mathbb{R}^n\) that contains the origin in its interior. Using the cone-volume functional \(U\), Lutwak, Yang and Zhang modified Schneider's projection problem. In the paper under review, the authors prove that the first variation of the cone-volume functional \(U\) with respect to the Minkowski sum exists and name it the mixed cone-volume functional. Namely, \[ U_1(K,L)=\frac{1}{n}\lim_{\lambda\to0^{+}}\frac{U(K+\lambda L)-U(K)}{t}, \] where \(K\) and \(L\) are convex bodies in \(\mathbb{R}^n\) with the origin in their interiors. Two basic inequalities involving \(U_1(K,L)\) and the first mixed volume \(V_1(K,L)\) are obtained: \[ \frac{n!}{n^n}\left(\frac{V(K)}{U(K)}\right)^{n-1} \leq\frac{U_1(K,L)}{V_1(K,L)}\leq \left(\frac{V(K)}{U(K)}\right)^{n-1}. \] In addition, the equality conditions are also provided. When \(L=K\), the first inequality becomes LYZ conjecture which was completely settled by \textit{K. J. Böröczky} and \textit{M. Henk} [Adv. Math. 286, 703--721 (2016; Zbl 1334.52003)].
scientific article; zbMATH DE number 7367366
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