Optimal problem for the $L_{p}$ mixed chord integral

In this paper we investigate the optimal problem for the L_{p} mixed chord integral. The existence of the L_{p} chord integral–Petty bodies is established, and the L_{p} geominimal chord integral is proposed and its properties, such as invariance under orthogonal matrices, homogeneity, isoperimetric-type inequalities and cyclic-type inequalities, which are provided as well.