The \(L_p\)-curvature images of convex bodies and \(L_p\)-projection bodies
This paper proves various inequalities of the \(L_p\)-Brunn--Minkowski theory. For example, if \(K,L\subset {\mathbb R}^n\) are convex bodies with \(0\) in the interior, \(L\) with positive continuous curvature function, then, for \(p\geq 1\), \[ V(K)^{n-p}\Omega_p(L)^{n+p} \leq \omega_n^n n^{n+p} V_p(L,\Pi_p K)^n. \] Here, \(V\) is the volume, \(\Omega_p\) is the \(L_p\)-affine surface area, \(V_p\) is the \(L_p\)-mixed volume, \(\Pi_p\) is the \(L_p\)-projection body, and \(\omega_n\) is the volume of the \(n\)-dimensional unit ball. Equality holds if and only if \(K\) and \(L\) are homothetic ellipsoids with centre \(0\). Special cases of this inequality, which for \(p=1\) is due to \textit{E. Lutwak} [J. Differ. Geom. 23, 1--13 (1986; Zbl 0592.52005)], include the \(L_p\)-Busemann--Petty centroid inequality and the \(L_p\) analog of the affine projection inequality. Further inequalities are proved, involving \(p\)-curvature images, mixed \(p\)-affine surface areas, \(L_p\)-projection bodies, and \(L_p\)-mixed volumes.
scientific article; zbMATH DE number 5354752
scientific article