On monotonicity properties of the \(L_{p}\)-centroid bodies (Q2853300)

\textit{E. Lutwak} [J. Differ. Geom. 38, No. 1, 131--150 (1993; Zbl 0788.52007)] introduced the notion of \(L_p\)-mixed quermassintegrals \(W_{p,i}(K,L)\) for any star bodies about the origin \(K\), \(L\in \mathbb{R}^n\), real \(p\geq 1\), and \(i=0,1, \ldots, n-1\). The notion of \(L_p\)-dual mixed quermassintegrals \(\widetilde W_{-p,i}(K,L)\) was defined by \textit{W. D. Wang} and \textit{G. S. Leng} [Indian J. Pure Appl. Math. 36, No. 4, 177--188 (2005; Zbl 1081.52009)] for any real \(i\neq n\).NEWLINENEWLINE The authors of the paper under review associate a new geometric body \(\Gamma _{p,i} K\), which they call \(L_p\)-centroid body, to each star-shaped body about the origin \(K\in \mathbb{R}^n\), real \(p\geq 1\), and real \(i\). It is proved that for any \(p\geq 1\), \(i=0,1, \ldots, n-1\), and real \(j\neq n, n+p\), if for any star body about the origin \(Q\) one has \(\widetilde W_{-p,j}(K,Q) \leq \widetilde W_{-p,j}(L,Q)\), then NEWLINE\[NEWLINE\frac{W_i(\Gamma _{p,j} K)^{-p/(n-1)}}{V(K)} \geq \frac{W_i(\Gamma _{p,j} L)^{-p/(n-1)}}{V(L)}\quad \mathrm{and}\quad \frac{\widetilde W_i(\Gamma^* _{p,j} K)^{p/(n-1)}}{V(K)} \geq \frac{\widetilde W_i(\Gamma^* _{p,j} L)^{p/(n-1)}}{V(L)},NEWLINE\]NEWLINE where \(V(K)\) and \(K^*\) denote the volume and the polar dual of \(K\), respectively. Moreover, equality holds in each relation precisely when \(K=L\).