The dual Orlicz Brunn-Minkowski inequality for the intersection body (Q2064774)

In this paper, the authors obtain a dual Orlicz-Brunn-Minkowski type inequality for the dual quermassintegrals of the intersection body in the \(n\)-dimensional Euclidean space. More precisely, they prove that if \(M\) and \(N\) are star bodies with positive radial function, \(1\leq j\leq n\) and \(1\leq i\leq n-1\), then \[1\leq\varphi\left(\dfrac{\widetilde{W}_{n-j}(I_iM)^{\frac{1}{ij}}}{\widetilde{W}_{n-j}\bigl(I_i(M\widetilde{+}_{\varphi}N)\bigr)^{\frac{1}{ij}}}, \dfrac{\widetilde{W}_{n-j}(I_iN)^{\frac{1}{ij}}}{\widetilde{W}_{n-j}\bigl(I_i(M\widetilde{+}_{\varphi}N)\bigr)^{\frac{1}{ij}}}\right)\] if \(\varphi:[0,\infty)^2\longrightarrow[0,\infty)\) is a continuous concave function, strictly increasing in each component, with \(\varphi(0)=0\) and such that \(\lim_{t\to\infty}\varphi(tx)=\infty\) for \(x\neq 0\). The inequality is reversed when \(\varphi:(0,\infty)^2\longrightarrow[0,\infty)\) is convex, strictly decreasing in each component, and satisfies that \(\lim_{t\to 0}\varphi(tx)=\infty\) and \(\lim_{t\to\infty}\varphi(tx)=0\). Here \(\widetilde{W}_j\) represents the \(j\)th dual quermassintegral and \(I_iM\) is the \(i\)th intersection body of \(M\), i.e., the star body whose radial function is given by \[\rho_{I_iM}(u)=\frac{1}{n-1}\int_{\mathbb{S}^{n-1\cap u^\bot}}\rho_M(v)^idv.\]