The $L_{p}$-Mixed Geominimal Surface Areas

In the paper, our main aim is to introduce a new concept and call it the $L_p$-mixed geominimal surface area $G_{p}(K_{1},\,\dots\,,K_{n})$ of $n$ convex bodies $K_{1},\,\dots\,,K_{n}$, which obeys the classical basic properties. The new affine geometric quantity in a special case yields Petty's geominimal surface area $G(K)$ of a convex body $K$, Lutwak's $p$-geominimal surface area $G_{p}(K)$ of $K$, and the newly established mixed geominimal surface area $G(K_{1},\,\dots\,,K_{n})$ of $n$ convex bodies $K_{1},\,\dots\,,K_{n}$. We establish some $L_{p}$-mixed geominimal surface area inequalities for the $L_p$-mixed geominimal surface area, whose some special cases are Petty's geominimal surface area inequality, Lutwak's $p$-geominimal surface area inequality, and some new mixed geominimal surface area inequalities.