The \(L^p\)-Busemann-Petty centroid inequality
The authors consider parallel chord movements of convex bodies; in such movements the chords of a convex body parallel to a fixed direction are shifted during a time \(t\) -- each chord with different speed -- with the only condition that the union of the shifted chords remains a convex body. These movements are particular cases of shadow systems and include as a special case the movement related to Steiner symmetrization. The authors prove that the family of all the \(p\)-centroid bodies of the parallel chord movements of a convex body along a fixed direction \(v\) is a shadow system along the same direction \(v\). They also prove that the volume function of the \(p\)-centroid body in such parallel chord movements is a strictly convex function unless the speed function of the movement is linear. As a consequence of this result they give an alternative proof of a result by Lutwak, Yang and Zhang characterizing the origin symmetric ellipsoids as the minimizers of the function \(\frac {V(\Gamma_p K)}{V(K)}\) where \(\Gamma_p K\) is the \(p\)-centroid body of \(K\).
scientific article; zbMATH DE number 1763529
scientific article